DESCRIPTION
Title of Invention: METHOD FoR CONTROLLING SECONDARY cooLINc oF
CONTINUOUS CASTING MACHINE AND DEVICE OF CONTROLLING SECONDARY
COOLING OF CONTINUOUS CASTING MACHINE
Technical Field
[0001] The present invention relates to a method for controlling secondary cooling of a
continuous casting machine and a device of controlling secondary cooling of a continuous
casting machine, which control a temperature distribution on a part of or a whole of a surface
of a slab in a casting direction or a width direction in a secondary coolin g zone of the
continuous casting machine.
Background Art
10002] For continuous casting of steel, for example, a vertical bending type casing
machine is employed. In this case, a slab extracted from a vertical mold is bent and is then
drawn with a certain curvature. Then, the slab is straightened in a straightening zone, and the
straight slab is withdrawn and cut. Meanwhile, a tensile stress is applied to a lower surface
of the slab in a curved portion of a strand (herein, the strand refers to a set of drawing parts
including a mold, a secondary cooling zone, and a group of rollers), and a tensile stress is
applied to an upper surface of the slab in the straightening zone. Therefore, when the slab
surface temperature is within a range called a stiffening area, a surface cracking defect called a
transverse crack may be generated. For this reason, it is necessary to appropriately set a
coolant amount distribution such that the slab surface temperature does not stay in the
stiffening area in the curved portion of the strand and the straightening zotre. The coolant
amount distribution may be appropriately set, for example, by setting the cooling zone water
distribution to a suitable value in advance through simulation or the like if the casting speed is
constant.
[0003] However, if arrival of the next ladle is delayed in the continuous casting, it is
necessary to wait for the arrival by reducing the casting speed under a predetermined level in
order not to halt the continuous casting. Therefore, it is necessary to change the casting
speed during the work. In this case, in a conventional cascading water amount control
technique, the water amount set for each zone in advance to match the casting speed is
changed in an interpolating manner dependìng on the changed casting speed. Therefore, a
time-dependent cooling hysteresis from a bath level of the mold to a position where cutting is
carried out may be disturbed, so that slab quality degradation such as a transverse surface
crack is generated.
[0004] In some cases, due to scale present on the slab surface or the like, a relationship
between the coolant amount and the heat transfer coefficient on the surface may vary from
those supposed through simulation in advance. Similarly, in this case, the slab surface
temperature may enter the stiffening area, and a transverse crack may be generated.
[0005] In such a problem, a so-called model-based predictive control method has been
disclosed. For example, a surface temperature control method is disclosed in patent
Literature 1, in which the extracted slab is tracked at regular intervals, and temperature
distributions of each tracking surface are sequentially calculated using a heat transfer model.
In addition, the model is corrected by reflecting a heat transfer coefficient learned from a
relationship between the actual measurement temperature and the calculated temperature in
exits of each zone obtained by segmenting the slab extraction path. Furthermore,
temperature distributions on each tracking surface at the temperature measurement points set
along the path are predicted on a regular time basis using the corrected model described above.
:''.lii-,iL :-; ììl;:;:;*:::r.j¿::;-;:::ì:ìi;-lt* : :,'.: :':: ::¿
Then, a feed-forward water amount obtained from a difference between a target temperature
and a predicted tempemture in those temperature measurement points and a feed-back water
amount obtained from a difference between the actual measurement temperature and the target
temperature are summed to a controlled water amount, and the controlled water amount is
sprayed to the slab.
Citation List
Patent Literature
[0006] PatentLiterature l: Jp 557-154364 A
Summary of Invention
Technical Problem
[0007] In the method of calculating the feed-forward water amount disclosed in patent
Literature l, temperatures of each tracking point of the cooling zone atthe timing when each
tracking point reaches the ternperature measurement point of the cooling zone exit are
predicted, a predicted water amount density is obtained such that the temperature prediction
value matches a target value when each tracking point reaches the temperature measurement
point. Furthermore, a weighted average of the predicted water amount density for all the
tracking surfaces of the corresponding cooling zone is set as the feed-forward water amount.
In this technique, the feed-forward. water amount is obtained sequentially from the mold-side
cooling zone, and the temperature distribution in the corresponding cooling zone is
recalculated using the feed-forward water amount obtained as described above. Then, the
recalculated temperature is set as an initial temperature in the cooling zone entrance
neighboring to the downstream side. These processes are repeated to determine the coolant
amount for all of the cooling zones. However, in this technique, even when the recalculated
temperature is set as the initial temperature at the cooling zone entrance neighboring to the
downstream side, the feed-forward water amount does not affect the temperature calculation
for the tracking points other than the cooling zone entrance neighboring to the downstream
side (the temperature calculation for a tracking point in any other coolin g zone placed in the
downstream side farther than the cooling zone neighboring to the downstream side from which
the recalculated temperature is obtained). Therefore, in the technique of patent Literafure l,
for the temperature prediction calculation, longer time is necessary until a change of the water
amount in the upstream side is appropriately reflected on the temperafure prediction
calculation. In some cases, a problem of hunting the water amount may arise. As a result,
accuracy may be easily degraded in control of the surface temperature of the entire slab to a
predetermined target temperature.
t0008] In view of the aforementioned problems, it is therefore an object of the present
invention to provide a method and device for controlling secondary cooling of a continuous
casting machine, capable of improving accuracy in control of the surface temperature of the
entire slab to a predetermined target temperature.
Solution to Problem
[0009] According to a f,trst aspect of the present invention, there is provided a method of
controlling secondary cooling of a continuous casting machine by dividing a secondary
cooling zone for cooling a slab extracted from a mold of the continuous casting machine into a
plurality of cooling zones in a casting direction of the slab and controlling a coolant amount
sprayed toward the slab in each cooling zone to control a slab surface temperature, the method
comprising: a slab surface temperature measurement step of measuring the slab surface
temperature at a predetermined temperature measurement point inside a strand during casting
of the slab; a casting speed recognition step of recognizing a casting speed of the continuous
casting machine; a tracking sutface setting step of setting a tracking surface as a target of
calculation for distributions of a temperature inside a cross-sectional profile of the slab, the
-,i :-::.:¡.:::ì:1i: :.'-1-â jj':i-ii
slab surface temperature, and a solid phase rate of the slab at a predetermined interval from a
position of a molten metal surface inside the mold at least to a cooling zone exit of a secondary
cooling control target; a slab target temperature setting step of setting a targeT value of the slab
surface temperature on the tracking surface; a temperature/solid-phase-rate estimation step of
calculating and updating distributions of the temperature inside the cross-sectional profile of
the slab perpendicular to the casting direction, the slab surface temperature, and the solid
phase rate of the slab using a heat transfer solidification model based on a heat transfer
equation whenever the tracking surface is shifted by a predetermined interval in the casting
direction of the slab as the casting progresses; a heat transfer coefficient estimation step of
calculating a heat transfer coeflicient on a slab surface used in the heat transfer solidification
model by using a casting condition including the coolant amount; a heat t¡ansfer/solidification
model parameter correction step of correcting a parameter for the casting condition in the heat
transfer solidification model using a difference between the slab surface temperature measured
in the slab surface temperature measurement step and the slab surface temperature estimated in
the temperature/solid-phase-rate estimation step; a prediction surface setting step of setting a
prediction surface for predicting distributions of the slab surface temperature at a subsequent
timing, the temperature inside the cross-sectional profile perpendicular to the casting direction,
and the solid phase rate of the slab at a predetermined constant interval in the casting direction
out of a set of tracking surfaces set in the tracking surface setting step; a prediction step of,
under assumption that the casting speed is not changed from the current timing while an
arbitrary prediction surface is shifted to a prediction surface position neighboring to its
downstream side from the current timing as casting progresses, repeatedly predicting and
updating distributions of the slab surface temperature, the temperature inside the
cross-sectional profile of the slab perpendicular to the casting direction, and the solid phase
rate of the slab when the prediction surface reaches a prediction surface position at the interval
used in the prediction surface setting step using the heat transfer solidification model; an
expected temperature influence coeffrcient prediction step of, under assumption that the
casting speed is not changed from the current timing whenever the arbitrary prediction surface
is shifted to a prediction surface position neighboring to its downstream side from the cur¡ent
timing as the casting progresses, predicting the slab surface temperature in each tracking
surface position in the middle of a path to the prediction surface position by assuming that the
coolant arnount of each cooling zone is changed in a step function shape, obtaining a deviation
between the predicted slab surface temperature and the slab surface temperature predicted in
the prediction step, and obtaining a change influence coefficient for the coolant amount that
changes in a step function shape using the deviation; a slab surface reference temperature
calculation'step of calculating a reference target temperature determined depending on time,
the reference target temperature being between a target value of the slab surface temperafure
set in the slab target temperature setting step and a prediction value of the slab surface
temperature at the timing when the prediction surface predicted in the expected temperature
influence coefficient prediction step reaches the prediction surface position; an optimization
problem coefficient matrix calculation step of setting the coolant amount of each cooling zone
at the current timing as a determinant, calculating an expected temperature influence
coefficient in each prediction surface position where each prediction surface of each of the
prediction step and the expected temperature influence coefficient prediction step passes, and a
deviation between the reference target temperatures calculated in the slab surface reference
temperature calculation step and the slab surface temperature predicted in the prediction step,
regarding an optimization problem for minimizing a sum of the deviations calculated for each
prediction surface as a quadratic programming problem, and calculating a coefficient matrix
for the determinant in the quadratic programming problem; an optmization problem solution
step of numerically calculating the quadratic programrning problem to obtain an optimum
value of a change amount of the coolant amount that changes in a step function shape at the
current timing; and a coolant amount change step of adding the optimum value to the coolant
amount of a current cooling zone to change the coolant amount, wherein, by repeatedly
changing the coolant amount in the coolant amount change step, the slab surface temperature
of the prediction surface in the prediction surface position is controlled to the target value of
the slab surface temperature set in the slab target temperature setting step while each tracking
surface at an arbitrary timing during the casting is shifted to a coolin g zoîe exit as the
secondary cooling control target.
[0010] According to a second aspect of the present invention, there is provided a device
for controlling secondary cooling of a continuous casting machine by dividing a secondary
cooling zone for cooling a slab extracted from a mold of the continuous casting machine into a
plurality of cooling zones in a casting direction of the slab and controlling a coolant amount
sprayed toward the slab in each cooling zone to control a slab surface temperature, the device
comprising: a slab surface temperature measurement unit configured to measure the slab
surface temperature at a predetermined temperature measurement point inside a strand during
casting ofthe slab; a casting speed recognition unit configured to recognize a casting speed of
the continuous casting machine; a tracking surface setting unit configured to set a tracking
surface as a target of calculation for distributions of a temperature inside a cross-sectional
profile of the slab, the slab surface temperature, and a solid phase rate of the slab at a
predetermined interval from a position of a molten metal surface inside the mold at least to a
cooling zone exit of a secondary cooling control target; a slab target temperature setting unit
configured to set a target value of the slab surface temperature on the kacking surface; a
temperature/solid-phase-rate estimation unit configured to calculate and update distributions of
the ternperature inside the cross-sectional profile of the slab perpendicular to the casting
direction, the slab surface temperature, and the solid phase rate of the slab using a heat transfer
iaì
iii
solidif,rcation model based on a heat transfer equation whenever the tracking surface is shifted
by a predetermined interval in the casting direction of the slab as the casting progresses; a heat
transfer coeffrcient estimation unit configured to calculate a heat transfer coefficient on a slab
surface used in the heat transfer solidification model by using a casting condition including the
coolant amount; a heat transfer/solidification model parameter correction unit configured to
correct a parameter for the casting condition in the heat transfer solidification model using a
difference between the slab surface temperature measured by the slab surface temperature
measurement unit and the slab surface temperature estimated by the
temperature/solid-phase-rate estimation unit; a prediction surface setting unìt configured to set
a prediction surface for predicting distributions of the slab surface temperature at a subsequent
timing, the temperature inside the cross-sectional profile perpendicular to the casting direction,
and the solid phase rate of the slab at a predetermined constant interval in the casting direction
out of a set of tracking surfaces set by the tracking surface setting unit; a prediction unit
configured to, under assumption that the casting speed is not changed from the cur¡ent timing
while an arbitrary prediction surface is shifted to a prediction surface position neighboring to
its downstream side from the current timing as casting progresses, repeatedly predict and
update distributions of the slab surface temperature, the temperature inside the cross-sectional
profile of the slab perpendicular to the casting direction, and the solid phase rate of the slab
when the prediction surface reaches a prediction surface position at the interval used by the
prediction surface setting unit using the heat transfer solidification model; an expected
temperature influence coefficient prediction unit configured to, under assumption that the
casting speed is not changed from the current timing whenever the arbitrary prediction surface
is shifted to a prediction surface position neighboring to its downstream side from the current
timing as the casting progresses, predict the slab surface temperature in each tracking surface
position in the middle of a path to the prediction surface position by assuming that the coolant
;^-_i;:i;;,:_i:i:::::ri;¡::::i:Jr-ir::i,l::: ì:'l:ì-i-:i-ri= : :: : :-:alì':;:*J;È;t ia:liù,
amount of each cooling zone is changed in a step function shape, obtain a deviation between
the predicted slab surface temperature and the slab surface temperature predicted by the
prediction unit, and obtain a change influence coefficient for the coolant amount that changes
in a step function shape using the deviation; a slab surface reference temperature calculation
unit configured to calculate a reference target temperature determined depending on time, the
reference target temperature being between a target value of the slab surface temperature set
by the slab target temperature setting unit and a prediction value of the slab surface
temperature at the timing when the prediction surface predicted by the expected temperatue
influence coefftcient prediction unit reaches the prediction surface position; an optimization
problem coefficient matrix calculation unit configured to set the coolant amount of each
cooling zone at the cur¡ent timing as a determinant, calculate an expected temperature
influence coefficient in each prediction surface position where each prediction surface of each
of the prediction unit and the expected temperature influence coefflrcient prediction unit passes,
and a deviation between the reference target temperatures calculated by the slab surface
reference temperature calculation unit and the slab surface temperature predicted by the
prediction unit, regarding an optimization problem for minimizing a sum of the deviations
calculated for each prediction surface as a quadratic programming problem, and calcul ating a
coefficient matrix for the determinant in the quadratic programming problem; an optimization
problem solving unit configured to numerically calculate the quadratic programming problem
to obtain an optimum value of a change amount of the coolant amount that changes in a step
function shape at the current timing; and a coolant amount change unit configured to add the
optimum value to the coolant amount of a current cooling zone to change the coolant amount,
wherein, by repeatedly changing the coolant amount using the coolant amount change unit, the
slab surface temperature of the prediction surface in the prediction surface position is
controlled to the target value of the slab surface temperature set by the slab target temperature
10
setting unit v/hile each tracking surface at an arbitrary timing during the casting is shifted to a
cooling zone exit as the secondary cooling control target.
Advantageous Effects of Invention
[0011] According to the present invention, it is possible to provide a method and device
for controlling secondary cooling of a continuous casting machine, capable of controlling the
surface temperature of the entire slab to a predetermined target temperature at all times. As a
result, it is possible to control the surface temperature such that a stiffening area of steel is
avoided in a bendin g zoîe or a straightening zone of the continuous casting machine even at
any casting speed or even when a casting speed is changed during casting. Therefore,
according to the present invention, it is possible to manufacture a slab having no defect due to
a surface crack.
Brief Description of Drawings
[0012] FIG. I is a diagram illustrating a continuous casting machine 9 and a cooling
control device 10;
FIG. 2 is a diagram illustrating division of a slab cross-sectional profile perpendicular to
a casing direction and an exemplary cell;
FIG. 3 is a diagram illustrating a cooling control method according to the present
invention;
FIG. 4 is a diagram illustrating a relationship befween a position of a tracking surface for
evaluating a surface temperature and a relative timing for predicting the surface temperature
while each prediction surface is shifted to a prediction surface position neighboring to its
downstream side;
FIG. 5 is a block diagram illustrating a relationship between each unit of the cooling
confrol device 10 and transmitted information;
ij
I7
FIG. 6A is a diagram illustrating a relationship between a slab width center surface
temperature at each cooling zone exit and time when a cooling control method according to
the present invention is applied to a case where a casting speed is reduced;
FIG. 68 is a diagram illustrating a relationship between a coolant amount at each
cooling zone and time when the cooling control method according to the present invention is
applied to a case where the casting speed is reduced;
FIG. 6C is a diagram illustrating a relationship between the slab width center surface
temperafure at each cooling zone exit and time when the cooling control method according to
the present invention is applied to a case where the casting speed is reduced;
FIG. 6D is a diagram illustrating a relationship between the coolant amount at each
cooling zone and time when the cooling control method according to the present invention is
applied to a case where the casting speed is reduced;
FIG. 6E is a diagram illustrating a relationship between the casting speed and time when
the cooling control method according to the present invention is applied to a case where the
casting speed is reduced;
FIG. 7A is a diagrarn illustrating a relationship between the slab width center surface
temperature at each cooling zone exit and time when a conventional cascading water amount
control technique is applied to a case where the casting speed is reduced;
FIG. 78 is a diagram illustrating a relationship between the coolant amount at each
cooling zone and time when the conventional cascading water amount control technique is
applied to a case where the casting speed is reduced;
FIG. 7C is a diagram illustrating a relationship between the slab width center surface
temperature at each cooling zone exit and time when the conventional cascading water amount
control technique is applied to a case where the casting speed is reduced;
FIG. 7D is a diagram illustrating a relationship between the coolant amount at each
72
cooling zone and time when the conventional cascading water amount control technique is
applied to a case where the casting speed is reduced;
FIG. 7E is a diagram illustrating a relationship between the casting speed and time when
the conventional cascading water amount control technique is applied to a case where the
casting speed is reduced;
FIG' 8A is a diagram illustrating a relationship between an actual measurement value of
the slab surface temperature , a fargeltemperature, and time when an exit target temperature of
a third cooling zone is changed during casting, and the cooling control method according to
the present invention is applied to a surface temperature control by controlling the coolant
amount;
FIG' 8B is a diagram illustrating a relationship between the coolant amount and time
when the exit target temperature of the third coolingzone is changed during casting, and the
cooling control method according to the present invention is applied to a surface temperature
control by controlling the coolant amount;
FIG' 8C is a diagram illustrating a relationship between the casting speed and time when
the exit target temperature of the third cooling zone is changed during casting, and the cooling
control method according to the present invention is applied to a surface temperature control
by controlling the coolant amount;
FIG. 9A is a diagram illustrating a relationship between an actual measurement value of
the slab surface temperature, the target temperature, and time when a spray heat transfer
coeffrcient of a fouth cooling zone is reduced, and the cooling control method according to
the present invention is applied to a slab surface temperature control by controlling the coolant
amount;
FIG. 98 is a diagram illustrating a relationshþ between the coolant amount and time
when the spray heat transfer coeff,tcient of the fourth cooling zone is reduced, and the cooling
13
control method according to the present invention is applied to a slab surface temperature
control by controlling the coolant amount; and
FIG. 9C is a diagram illustrating a relationship befween the casting speed and time when
the spray heat transfer coeffìcient of the fourth cooling zone is reduced, and the cooling
control method according to the present invention is applied to a slab surface temperature
control by controlling the coolant amount.
Description of Embodiments
[0013] Embodiments of the present invention will now be described. Note that the
embodiments described below are just for illustrative purposes and do not limit the scope of
the invention.
[0014] FIG. 1 is a diagram illustrating a continuous casting machine 9 according to the
presentinvention and a device 10 of controlling secondary cooling of the continuous casting
machine according to the present invention (hereinafter, also referred to as a,,cooling control
device"). In FIG' 1, the continuous casting machine 9 and the cooling control device 10 are
illustrated in simplified forms.
In the continuous casting machine 9 according to the present invention, a strand having
a solidified outer side is nipped and supported by pairs of rolls, and the strand is extracted
from a mold I at a predetermined extraction speed (casting speed) using pinch rolls connected
to a driver. The reference numeral "4" denotes a molten steel meniscus. A spray nozzle of a
mist spray 2 (or spray 2) for spraying a coolant toward the slab 5 is placed between
neighboring support rolls arranged at predetermined intervals in the casting direction. A flow
rate of the sprayed coolant is controlled by a flow rate control valve 3 provided in a coolant
pipe. An opening level of the flow rate control valve 3 is adjusted on the basis of a water
amount instruction value from the cooling control device 10. Since the coolant pipes are
I4
installed to match cooling zones obtained by segmenting a length of the slab 5 in the casting
direction into a plurality of zones (the cooling zones segmented by cooling zone boundary
lines 6), a coolant amount distribution of the casting direction of the strand is controlled on a
cooling zone basis. In the following description, the cooling zones will be sequentially
numbered by the first, second, ..., cooling zone from the cooling zone immediately under the
mold. Note that the "casting direction" refers to a longitudinal direction of the slab.
[001 5] Distributions of the temperature and the solid phase rate of the slab 5 in the strand
are calculated by setting cross-sectional profiles perpendicularly to the slab 5 at calculation
points established at regular intervals in the casting direction from the molten metal surface
inside the mold to a final rolls exit side and solving a heat transfer equation obtained by
discretizing distributions of the temperature and the solid phase rate of each cross-sectional
profile under a boundary condition of a heat transfer coefficient by reflecting cooling
conditions at each calculation points. As an initial condition of the heat transfer equation,
calculation results for the temperature and the solid phase rate of the cross-sectional profile
neighboring to the upstream side of the cross-sectional profile at the calculation target position
aÍe established' In addition, the calculation is repeated from the calculation point
neighboring to the upstream side to the target calculation position until the cross-sectional
profile is shifted by the slab extraction. As a result, it is possible to calculate the temperature
and the solid phase rate on the entire slab.
[0016] For the discretization of the heat transfer equation, for example , a 2-dimensional
rectangular cell model illustrated in FIG. 2 is employed. At each cell (i, j), a temperature T¡,
enthalpy per unit mass H¡, and a solid phase rate per unit mass f¡ are set as variables, and
physical constants of each cell (i, j) are expressed as a density p¡, a specific heat C¡, and a
thermal conductivity À¡ in consideration of temperature dependency. In this case, a
relationship between the enthaþy H¡, the temperature T¡, and the solid phase rate f¡ can be
15
expressed as Equation (1).
[0017]
[Math. 1]
Hr= prCrTu + pu (t-.f)Lu Equation (1)
l00l8l Temporal changes of the distributions of the enthalpy H¡ and the solid phase rate
f¡ on the cross-sectional profile extracted from a position z ro aposition ztLz ofthe casting
direction during time interval At can be expressed using discrete heat transfer equations (2),
(4), and (7), an initial condition equation (3), and boundary condition equations (5), (6), (g),
and (9). In the following equations, a superscript "2" denotes a position of the casting
direction, and a molten metal surface position inside a mold is set to ,,2 : 0.,, The time
interval At of the heat transfer equation is transformed to "^t : Lz/v(t-l),' using a profile unit
length Âz of the casting direction and a casting speed v(t-l) at the timing (t-l). Heat from the
slab surface is set by reflecting a boundary condition considering a difference of the cooling
method depending on a cross-sectional position of the casting direction such as cooling caused
by the coolant sprayed to the slab 5, contact with the rolls, and radiation, Here, the heat from
the slab surface is represented by a heat transfer coefficient K* or K, in a linear equation
regarding a difference between an ambient temperature Tr and a surface temperature T¡, as
expressed in Equations (5) and (8).
[001e]
fMath.2]
prcrfuï*Q +t)- u", Ø\ln
= 1 f Ax, {- øî, p,,O * qí-v,.,Q)\ + t I Ly, {- øí,¡ u¡, Q) + q;.
-, t, Q)} Equation (2)
[0020]
I6
lij
;,?
l-:l
dl.
-*
[Math.3]
Initial condition: temperature Tt'(t), solid phase rate fi¡" (i = 1, . . ., I+1, j = l, . .., J+1)
Equation (3)
[0021] In Equation (2) described above, "{i*u2,¡" denotes a heat flux from the cell (i, j)
to the cell (i+1, j) in the slab width direction at the casting-direction position (z-1) and can be
expressed as Equation (4) described below by setting the inside of the slab width direction to
"i: 2, ..., f ." In the following description, the slab width direction will be simply refered to
as a "width direction" as well.
100221
fMath.4]
Qí*t¡2.¡ = - 1i+r¡2.¡ Q:,., - r)l * Equation (4)
[0023] Note that "Ljj" in Equation (1) described above denotes a solidification latent heat
À"¡+t/2, j: (l"i+r, ¡+ Xi)/2 in the cell (i, ¡;, "axi" in Equation (2) described above denotes a
distance from the cell (i-I/2,j) to the cell (i+l/2,j), and "Âyi" in Equation (2) described above
denotes a distance from the cell (i, j-l/2) to the cell (i, j+l/2). In addition, a boundary
condition of the width direction is expressed as Equation (5) described below using the heat
transfer coefticient IÇ at the casting-direction position (z-1) and the ambient temperature Ts
assuming that a short side surface is set to "i : 1."
100241
fMath.5J
Qí-t¡2.¡ = *,þ; -,)
[0025] If a center line of the width direction is set to "i
condition is assumed as expressed in Equation (6) described
Equation (5)
: I*1," a symmetrical boundary
below on the center line of the
1'1
:f
Ê
t
:ll
,.i
5
width direction.
100261
[Math.6]
Qí*ttz.¡ =0 Equation(6)
100271 In Equation (2) described above, "l'i, j*v2" denotes a heat flux in a thickness
direction from the cell (i, j) to the cell (i, j+1) and can be expressed as Equation (7) described
below by assuming that the inside of the thickness direction is set to " j : 2, . . ., J."
[0028]
fMath.7]
Qí.¡ *t¡z = - l,,, rlrQÍ.,*, - rí)lny Equation (7)
[0029] Note that a relationship "X¡, j*ttz: (Ài, j*r*Àü)12" is established. In Equation (7)
described above, "Ay" denotes a distance from the cell (i, j) to the cell (i, j+l). In addition, a
boundary condition of the thickness direction is expressed as Equation (8) described below
using the heat transfer coefficient Ç at a casting-direction position (z-I) and an ambient
temperature Te by assuming that a long side surface is set to 'T = 1."
[0030]
[Formula 8]
Qí..¡-r¡r: Xr@, -fr) Equation (8)
[0031] If a thickness center line is set to 'T:J+l," a symmetrical boundary condition
expressed as Equation (9) described below is presumed on a center line of the
thickness-direction.
[0032]
[Math.9]
a()
Equation (9)
i":j'::1!r:;:-r.;--,:---.--.''r--:--:ù:-:.:;:iü1::¿ii:;:aà:!;a:i:;;t;:;
Qí,¡*tlz =0 fr (e)
[0033] If a perfect liquid phase 4f z+^z: 0" or a perfect solid phase "fijz+^z: l" is
established after the enthalpy ]H¡j'** at the casting-direction positi on (z+Lz) is calculated, a
temperature Trj'*o' is obtained by applying each value to Equation (1) described above.
Meanwhile, in the case of "0 < ¡t''*t' ( 1," the temperature Ti¡z+Lz matches a liquidus
temperature Tl(Cr.) (where "Ck" denotes a concentration of the solute component k) expressed
in a phase diagram determined by a solute concentration in a liquid phase. As known in the
art as i'Scheil Equation," a solute concentration of the liquid phase depends on the solid phase
rate. Therefore, using a model expressed in Equation (10) described below, the solid phase
raïe fi¡'+Mand the temperature T¡z+Lz are obtained as a solution of a simultaneous equation
between Equations (10) and (l).
[0034]
[Math. 10]
T;** ,
[003s]
=r,{cr(f;.Ù Equation (10)
If the heat flux emitted from the slab surface where the coolant sprayed from the
mist spray 2 collides is expressed as Equation (11) described below, the heat transfer
coefficient k is obtained using Equation (12) as follows.
[0036]
[Math. 11]
q = cTs" D#vl
100371
fMath. 12]
¡=q¡(r, -ro)
Equation (11)
Equation (12)
where "Tg" denotes a surface temperature [oC], "D*" denotes a surface water amount density
19
lu*21, and "vu" denotes a mist spray air flow velocity [m/s], and .,cr{,,,, ..þ,)' ..y,))
aÍrd ,,c,, ate
constants.
[0038] The cooling control device 10 obtaìns a prediction value of the slab surface
temperature at a temperahrre evaluation point using an extraction speed of the slab 5, a molten
steel temperature inside a tundish, and a coolant temperature. In addition, an optimum value
of the coolant amount for each cooling zone is calculated such that an evaluation function
determined by a deviation between the prediction value and a target value of the slab surface
temperature at a predetermined temperature evaluation point of each coolin g zone and the
coolant amount is minimized. In the method for controlling secondary cooling of the
continuous casting machine according to the present invention (hereinafter, also referred to as
a "cooling control method according to the present invention"), the slab surface temperature at
each tracking surface is controlled to the target value of the predetermined slab surface
temperature by repeating calculation described below within a single control cycle. The
cooling control method according to the present invention will now be described with
reference to FIG. 3 which illustrates the cooling control method according to the present
invention.
[0039] As illustrated in FIG. 3, the cooling control method according to the present
invention includes a slab surface temperature measurement step S1, a casting speed
recognition step 52, a tracking surface setting step 53, a slab target temperature setting step 54,
a temperature/solid-phase-rate estimation step 55, a heat transfer coeffìcient estimation step 56,
a heat transfer/solidification model parameter correction step 57, a prediction surface setting
step S8, a prediction step 59, an expected temperature influence coefficient prediction step Sl0,
a slab su¡face reference temperature calculation step Sll, an optimization problem coefficient
matrix calculation step S12, an optimization problem solution step S13, and a coolant amount
change step S14.
i!
$
20
.;il.i-.i1.ì::,,;l-1";.::::::!--i:i_:L*:raii1:iii:ì:;]l:.::i::-'r:i-i:i:i:r r:
[0040] In the slab surface temperature measurement step (hereinafter, also referred to as
"step Sl"), a slab surface temperature at a predetermined temperature measurement point on
the slab surface of the strand is measured using a slab surface temperature thennometer 7
during casting.
[0041] In the casting speed recognition step (hereinafter, also referred to as "step S2,,), a
slab extraction velocity (casting speed) of the continuous casting machine 9 is rneasured
sequentially by the casting speed measurement rolls 8 in order to recognize the casting speed.
Altematively, in step 52, the casting speed may be recognized, for example, by receiving data
on a setting value of the casting speed from an upper-layer computer (not shown) of the
cooling control device 10.
100421 In the tracking surface setting step (hereinafter, also referred to as "step S3,,), a
tracking surface as a target of calculating distributions of the temperature inside the slab
cross-sectional profile, the slab surface temperature, and the solid phase rate is set at
predetermined intervals within an area ranging from a position of the molten rnetal surface
inside the mold at least to the coolin g zone exit as a secondary cooling control target.
[0043] In the slab target temperature setting step (hereinafter, also referred to as ,,step
S4"), a target value of the slab surface temperature is determined on the tracking surface set in
step 53.
[0044] In the temperature/solid-phase-rate estimation step (hereinafter, also referred to as
"step S5'?), whenever the tracking surface set in step 53 is shifted by a predetermined interval
in the slab casting direction as the casting progresses, distributions of the temperature inside
the slab cross-sectional profile perpendicular to the casting direction, the slab surface
temperature, and the solid phase rate are calculated and updated using a heat transfer
solidification model based on aheat transfer equation.
In step 55, change amounts, from the previous control cycle, of the distributions of the
2I
temperature and the solid phase rate on cross-sectional profiles set perpendicularly to the slab
casting direction at predetermined intervals are calculated by solving the heat transfer equation
obtained by considering transformation heating generated when steel is solidified.
More specifically, assuming that the current timing is set to "1," aÍrd the Equations (2) to
(10) express variable relationship models between the timings "r" and"t-l," the distributions
of the temperature and the solid phase rate on the cross-sectional profile are calculated from
the calculation point neighboring to the molten metal surface inside the mold to the cooling
zone exilas a secondary cooling control target.
[0045] In the heat transfer coefficient estimation step (hereinafter, also referred to as
"step 56"), the heat transfer coefficients on the slab surface used in the heat transfer
solidification model (that is, the heat transfer coeffrcients used in Equations (5) and (8)
described above) are calculated using an estimation value of the heat transfer/solidification
model parameter at the current timing "t" and a casting condition such as the coolant amount
at the timing "t-1."
[0046] In the heat transfer solidification model parameter correction step (hereinafter,
also referred to as "step S7"), parameters for the casting condition in the heat transfer
solidification model are corrected using a difference between the slab surface temperature
measured in step sl and the slab surface temperature estimated in step s5.
100471 A parameter for the casting condition in the heat transfer solidification model are
corrected by multiplying a correction coefficient by a difference between the slab surface
temperature measured in step 51 and the estimation value of the slab surface temperature
estimated in step 55 and adding the multiplication result to the parameter for the casting
condition in the heat transfer solidification model as a model parameter correction amount. If
a plurality of measurement points for the slab surf,ace temperature (hereinafte¡ referred to as a
"temperature measurement point" or "temperature measurernent position") are provided, the
22
correction coefflicient is expressed as a matrix or a vector. The correction coeffìcient used in
the correction of the parameter for the casting condition in the heat transfer solidification
model is obtained in the following way for each estimation target parameter. Note that the
"parameters for the casting condition in the heat transfer solidification model,, refers to, for
example, the coefficient "c" in the right side of the heat flux modeling equation (ll), indices
for temperature etc., cr, B, and y, andthe like.
[0048] 1) The correction target parameter is set by slightly changing the current value.
2) For a cross-sectional profile placed in the temperature measurement position zu at the
current timing t, distributions of the temperature and the solid phase rate on the cross-sectional
profile in the position zt(rTa) at the timing (t-Ta) by reversely counting a predetermined time
Ta are set to initial values. In addition, when the parameter is slightly changed at the current
timing t, a temperature estimation value in the temperature measurement point is calculated by
applying a hysteresis of the cooling condition from the position zu(t-Ta) at the timing t-Ta to
the temperature rneasurement position zk at the current timing t and repeating the calculation
of Equations (2) to (10), The retro-counting range Ta may be set such that the correction
target parameter affects a state of the cross-sectional profile placed in the temperature
measurement position zç.
3) A linear relationship equation that represents a relationship between the correction
amount of each parameter and the temperature change amount is obtained in the following
sequence.
Assuming that the surface temperature estimation value calculated from Equation (2)
described above for the surface temperature Tk(t) estimated in step 55 is changed to .,T¡*AJ¡¡,,
when the parameter 0r is changed only by "Agr," the following Equation (13) can be obtained.
[004e]
[Math.l3]
¿J
Equation (13)
LTtt=AitLq ñ (r g)
where the estimation value of ,.A1i, in Equation (13) can be expressed as the following
Equation (14).
[0os0]
[Math. 14]
Ai,= LTuQ)lte, Equation (14)
It is noted that assuming "rqau," denotes a matrix Au of the (k)th row and the (l)th column,
a temperature change estimation value obtained by considering influence on the surface
temperature of the temperature measurement point for all of the correction target parameters
can be expressed as "A.AO" using a vector "40 : lA0r, A0z, ..., A0r]r where ,,Â01,, denotes the
(l)th element of this vector.
[0051] An optimum correction amount of the parameter is determined such that a vector
rga(t) representing a deviation V1(t) be¡¡¿een the temperature measurement values T"r(t) and
Tt(t) of each temperature measurement point as expressed in Equation (15) described below
can be most appropriately approximated to a temperature change AaÂQ obtained from the
corrected parameter considering a numerical calculation effor or dispersion of data.
[0052]
fMath. l5J
øiQ) =r; Ø-rr(ù Equarion (15)
[0053] That is, assuming that "ÂAu" denotes a matrix representing an error of each
element of the gain matrix *Aa,Ð a value capable of minimizing a solution of Equation (16) is
obtained as follows.
[00s4]
_ [Math. 16]
:=:::::J::: iJ::j::ii;i;-,
24
r = (lø" (ù - (¿' * u') nel'| Equation (16)
where "" denotes an expectation value of the variable .,x.,,
10055] A minimum value of "J" can be obtained analytically, and a parameter correction
amount A0(t) capable of minimizing"J" can be expressed as the following Equation (17).
[0056]
[Math. 17J
Le (ù = Q'Tn" * (t,t", na))-'¿" ø' (ù Equation (17)
where it is assumed "<ÂAa> : 0." Assuming that a correlation of each element of the gain
matrix is set to zero, the expectation value <ÂAurAAu> of the gain matrix can be expressed as
a matrix having variances of diagonal elements AAaii in the same diagonal positions.
Therefore, the value <^Aur^Au> is determined in advance on the basis of knowledge such as a
process.
The pararneter cor¡ection amount Â0(t) obtained as described above is added to the
current parameter.
[00s7]
fMath. 18]
0Q +r) = e(ù + neO Equation (18)
The result of the Equation ( 1 S) is used in calculation of the control setting amount at the
subsequent timings.
[0058] In the prediction surface setting step (hereinafter, also referred to as "step S8"), a
prediction surface for predicting distributions of the slab surface temperature, the temperature
inside the slab cross-sectional profile, and the solid phase rate in the subsequent timings is set
at regular intervals in a predetermined casting direction out of a set of the tracking surfaces set
25
:ì ìl:.;tr.:r:::.1111.::úì,;-i:::,Ê:¿ij:s. -i:lri.: -,i.;ì;;-;;;::.:
in step 53.
[0059] In the prediction step (hereinafter, also referred to as "step Sg'), assuming that the
casting speed is not changed from the current timing while an arbitrary prediction surface set
in step S8 is shifted to a position of the prediction surface neighboring to the downstream sicle
from the current timing as the casing progresses, distributions of the slab surface temperature,
the temperature inside the slab cross-sectional prof,rle, and the solid phase rate when each
prediction surface set in step S8 reaches a position ofthe prediction surface neighboring to the
downstream side are updated by repeatedly predicting them using the heat transfer
solidification model at each interval (heat transfer calculation interval) determined in step S8.
In step 59, distributions of the slab surface temperature, the temperature inside the slab
cross-sectional profile, and the solid phase rate arepredicted using the casting speed at the
current timing, the coolant amount of each cooling zone, and the parameter value of the heat
transfer solidification model corrected in step 57. As initial values for the prediction
calculation, distributions of the slab surface temperature, the temperature inside the slab
cross-sectional prof,rle, and the solid phase rate on each expected temperature prediction
surf,ace at the current timing t obtained in step 55 are used. Note ihat the "position of the
prediction surface" refers to a position of the prediction surface set in step Sg.
[0060] FIG. 4 illustrates a relationship between a position of the tracking surface for
evaluating the surface temperature and a relative timing for predicting the temperature while
each prediction surface set in step S8 is shifted to the prediction surface position neighboring
to the downstream side. In the following description, the position of the tracking surface will
be simply referred to as a "tracking surface position." In FIG. 4, The surface temperature is
predicted at the timing indicated b¡r a dot ,.r." A slope of the inclined line obtained by
linking a plurality of dots "." indicates a casting speed "v(t)" at the current timing .,1.,, In
step 59, a slab surface temperature prediction value on the tracking surface position ,,zi,,of the
26
:l: - i..;,- ";:-..-;;;i:;l-:;:,i:i tr:: J:ì;i;;:;i;;::a_i:rì;:rx-,::=::::i::.;
prediction surface "i" is denoted by a prediction temperature ,,Tpred¡.::
[0061] In the expected temperature influence coefficient prediction step (hereinafter, also
referred to as "step S10"), assurning that the casting speed is not changed from the current
timing whenever the prediction surface set in step S8 is shifted from the current timing to the
prediction surface position neighboring to the downstream side as the casting progresses, and
the coolant amount of each cooling zone is changed in a step function shape, the slab surface
temperatures in each tracking surface position in the middle of a path to the prediction surface
position neighboring to the downstream side are predicted. In addition, a deviation befween
this predicted slab surface temperature and the slab surface temperature predicted in step 59 is
obtained, and a change influence coefficient (also referred to as an "expected temperature
influence coefficient" for the coolant amount changing in a step function shape is obtained
using this deviation.
In step S10, for each cooling zone k, the slab surface temperature Tk¡ when the
prediction surface "i" reaches the prediction surface position z¡ neighboring to the downstream
side in the casting direction is predicted by assuming that each coolant amount q.(t) is changed
by an amount Aqt in a step shape at the current timing "t". In addition, a relationship
between the amount Aqr and a deviation "ÂTk¡(t) : 1kÙ-1nred-,,between the predicted slab
surface temperature Tk¡ and the prediction temperature Tp'"dù obtained in step 59 is expressed
as the following Equation (19).
[0062]
[Math. 19J
^r; = MI^qo Equation (19)
In Equation (19), the coefficient Mk¡ denotes the expected temperature influence
coefficient. In step S10, for each prediction surface, a surface temperature change gain
matrix "M" is calculated, in which the expected temperature influence coeffrcients Mk¡ are
2l
i::.1
arrangecl in'J" rows and "k" columns.
[0063] In the slab surface reference temperature calculation step (hereinafter, also
referred to as "step S11"), a reference target ternperafure as an intermediate target value
deteminecl depending on time (a temperature gradually approaching the target value of the
slab suface temperature set in step 54 whenever the prediction of step S10 is repeated) is
calculatecl. The reference talget temperature is set between the target value of the slab
surface temperature set in step 54 and the prediction value of the slab surface temperature
obtained when the prediction surface predicted in step S10 reaches the prediction surface
position.
ln step S11, for example, a reference target temperature T'el¡¡ at a temperafure evaluation
point z¡ of a cross-sectional profile at an entrance of the (i)th cooling zorrc aïthe current timing
can be set as a temperature between the prediction temperature lnted¡¡ and the target
temperature 1ts\ divided at a ratio determined by an exponential function of time t¡¡ as
expressed in the following Equation (20). In step Sl1, a reference target temperature track
T''''ii(t) expressed as a time function is obtained.
[0064]
IMath.20]
T,j"I =qrxr +exp(-t,,f r,)(r,l'*u Equation (20)
where "T1' denotes a time constant corresponding to a predetermined decay parameter.
[0065] In the optimization problem'.eoefficient matrix calculation step (hereinafter, also
referred to as "step S12"), the coolant amount of each cooling zone at the current timing "t" is
set as a determinant, and the expected temperature influence coefficients in each prediction
surface position where each prediction surface passes in each of steps'S9 and S10, and a
deviation between the reference target temperature and the prediction temperature on the slab
-r;Ù
surface are calculated. In addition, an optimization problem for minimizing a total sum of
the calculatecl cleviation for each prediction surface is establishecl as a quadratic programrning
problem, and a coefficient matrix for the determinant in the quadratic programming problem is
calculated.
In step 572, a weighted sum of squares of the difference between the slab surface
temperature response "1nred¡¡(t)+ATü(t)" in each evaluation position z¡ at the evaluation timing
"t" of step S11 and the reference target temperature track "T'"rì¡(t)" and a sum of squares of the
change step width Âqr of the coolant amount in each cooling zone are totally summed. In
acldition, this total sum is set as an evaluation function, and a value "Âq: [Aqr, Âqr, ..., Âq*]t
is obtained to minimize this evaluation function. The evaluation function is expressed as the
following Equation (21).
t00661
fMath.21]
J =>@,,,"u * LT _ T,,r)t g(Tn,*, + LT -T,Ð+ Lq't RLq Equation (21)
¡
where "TPred ," cclrer',rr and "ATi" are expressed as Equatio ns (22), (23), and.(24), respectively.
t00671
lMath.22l
T,*" = hï"' T!'*' Tl*'7'
. 100681
fMath.23]
T*, =T1T' T;T ,:A,
[006e]
Equation(22)
';-
29
Equation (23)
:ú::i::itJ:i.:-;i¿ì::ì
fMath.24]
nq =fnT,, LT. LT,.,f'
A term regarcling the temperature deviation in the evaluation function can be substituted
with the following Equation (25) using the gain rnatrjx obtained in step SiO. In aclclition, by
excluding a term impertinent to the change step width Aqr, of the coolant amount, the
minirnization of the evaluation function is equivalent to minirnization of a ftinction J' as
expressed in the following Equation (26).
[0070]
fMath.25]
^Tþ)
= M,Q)nq Equation (25)
Equation (24)
[0071]
[Math.26]
J' = > ín'r,Q)' eu,nq +z@n"''r Q) -r'"t(ù'oM,Q) Lq\ + a'qr RLq
= o¿ {1, Q)' eu,* n}aø +Z(rn'*t (ù-T'' Ø)'o>,u |¡tn Equation (26)
lO07Zl The minimizalinnof the function J' is a quadratic programming problem having a
cletenninant Aq. The matrix Q is a nonnegative definite "IxI" matrix, and the matrix R is a
positive definite "KxK" matrix. For example, the matrix Q is a diagonal matrix having
diagonal elements which are non-negative constants, and the matrix R is a diagonal matrix
having diagonal elements which are positive constants. In addition, it is possible to reflect a
phVsical constraint on the mist spray 2 by adding a constraint based on upper'and lower
limitations of the change step width of the coolant amount or upper and lower limitations of
the coolant amount.
t0073] In the optimization problem solution step (hereinafter, also referred to as "step
S13,'), an optimum value Aq* of Aq at the current timing is obtained by mathematically
30
solving the quadratic programming problem of step S12. Since the quadratic programming
problern is a convex quadratic programming problern, the optimurn solution Äq* is obtained
from the following Equation (21) if there is no constraint in Aq. Otherwise, if there is a
constraint in the change amount Àq, the optimum solution Aq* can be easily obtainecl by
applying an active set method or the like.
100741
lMath.27)
o¡ ={1*,(ù' ou,. Rf Ðm Ø'g(r,,*,Q) -T,",(ù) Equation (27)
[0075] In the coolant amount change step (hereinafter, also referred to as "step S14?'), the
optimum solution Aq* obtained in step S13 is added to the coolantamount q(t) of the current
cooling zone, so that Equation (28) is obtained as follows.
[0076]
[Math.28]
qQ +t) - q(ù+
^q.
Equation (28)
The coolant amount q(t+l) obtained in this manner is used in the next control cycle.
100771 Using the cooling control method including steps S1 to S14 according to the
present invehtion, it is possible to directly reflect influence of a change of the coolant amount
even on positions of the tracking surface used to evaluate the surface temperature other than
the cooling zone entrance neighboring to the downstream side in the casting direction.
Therefore, it is p.ossible to control.the surface temperature on the entire slab such that it
matches a predetermined target temperature at all times. As a result, using the sesling
control method according to the present invention, it is possible to improve accuracy for
controlling the surface temperature of the entire slab to a predetermined target temperature.
By controlling the surface temperature of the entire slab to the target temperature with high
31
'Ìi
accuracy, it is possible to control the surface temperature such that it can avoid a steel
stiffening area in a bending or straightening zone of the continuous casting machine even in
any casting speed or even when the casting speed is changed during the casting. Accordingly,
it is possible to manufacture a slab having no clefect due to a surface crack.
[0078] The cooling control method according to the present invention described above
can be implemented, for example, using the cooling control device 10 illustrated in FIG. 5.
As illustrated in FIGS. 1 and 5, the cooling control device 10 includes a slab surface
temperature thermometer 7 serving as a slab surface temperature measurement unit 7, a casting
speed measurement roll 8 serving as a casting speecl recognition unit 8, a tracking surface
setting unit 10a, a slab target temperature setting unit lOb, a temperature/solid-phase-rate
estimation unit 10c, a heat transfer coefficient estimation unit 10d, a heat transfer/solidification
model parameter correction unit 10e, a prediction surface setting unit 10f, a prediction unit 10g,
an expected temperature influence coefficient prediction unit lOh, a slab surface reference
temperature calculation unit 10i, an optimization problem coefficient matrix calculation unit
10j, an optimization problem solving unit 10k, and a coolant amount change unit 101. As
described above, the slab surface temperature thermometer 7 is used in step S1, and the casting
speed measurement roll 8 is used in step 52. In addition, the tracking surface setting unit IOa
is used to perform step 53, the slab target temperature setting unit lOb is used to perfonn step
54, the temperature/solid-phase-rate estimation unit 10c is used to perform step 55, the heat
transfer coefficient estimation unit 10d is used to perform step 56, and thè heat
transfer/solidification model parameter correction unit 10e is used to perforrn step 57. In
addition, the prediction surface setting unit 10f is used to perfonn step S8, the prediction unit
10g is used to perform step 39, the expected temperature influence coeffrcient prediction unit
10h is used to perform step S10, the slab surface reference temperature calculation unit 10i is
used to perform step 511, the optimization problem coeffrcient matrix calculation unit 10j is
)z
used to perfonn step S12, the optimization problern solving unit 10k is used to perform step
S13, ancl the coolant amount change unit 101 is usecl to perform step S14. Therefore, using
the cooling control device 10, it is possible to implernent the cooling control method according
to the present invention. According to the present invention, it is possible to provide a device
of controlling seconclary cooling of a continuous casting machine, by which the surface
temperature of the entire slab can be controlled such that it rnatches a predetermined target
temperature at all times.
Examples
t0079] Exarnples of the present invention applied to a first cooling zone clirectly under
the mold exit to a final tenth cooling zone in a continuous casting machine for a slab will now
be described
Assuming that the casting speed is constanl the temperature target value was obtained
using a slab surface temperature calculation value of the tracking surface position obtained by
calculating strand heat transfer solidification when each coolin g zone water amount was
optimized. The continuous casting machine used in Examples was a continuous casting
machine for a slab having a width of 2300 rnm, a thickness of 300 mm, and a length of 28.5 m
from the lneniscus position of the mold to the secondary cooling zone exit. In examples, an
update interval of the heat transfer calculation was set to 25 mm, a tracking surface interval
was set to I25 mm, and an expected temperature prediction interval was set to 1.25 m. On
the tracking surface, a cross-sectional profile of the slâb was divided into. four quarter profiles
with respect to a long-side center line and short-side center line (see FIG. 2). The slab was
divided into twenty segments in the thickness direction and into forty segments in the width
direction in order to calculate the heat transfer solidification model.
Note that the slab surface temperature \Mas measured at a distance of 5.25 m from the
33
meniscus of the fourth cooling zone exit side in the center of the long-side surface using a
radiation thermometer.
[0080]
Example 1
In Example 1, the cooling control method according to the present invention was
applied to a case where the slab injection speed was reduced by 25% during casting. In
Example 1, a relationship befween the surface temperature at the slab width center jn each
cooling zone exit and time is illustrated in FiGS. 6A and 6C, a relationship between the
coolant amount of each cooling zone and time is illustrated in FIGS. 6B and åD, and a
relationship between the casting speed and time is illustrated in FIG. 68. When the casting
speed was abruptly reduced from 0.8 m/min to 0.6 m/min and was then recovered to 0.8
m/min after 5 minutes, a square root of the difference between the slab surface temperature of
each coolin g zone exit and the target temperature in Example I was betwe en lZ"C and 18"C.
Meán*hile, a water amount cascading control was applied to a case where the slab
injection speed during casting was reduced by 25% (as a comparative example). A result of
the comparative example is illustrated in FIGS 7A to 7E. Specifiòaily, in the comparative
example, a relationship between the slab width center surface temperature at each cooling zone
exit and time is illustrated in FIGS. 7A and 7C, and a relationship between the coolant amount
of each cooling zone and time is illustrated in FIGS. 7B and 7D. In addition, a relationship
between the casting speed and time is illustrated in FIG. 7E. ln the comparative example,
even when the casting speed was changed under the same condition as t{at of Example l, .the
square root of the diff,erence between the slab surface temperature of each cooling zone exit
and the target temperature was between lToC and24"C. As illustrated in FIGS. 6A to 6E and
TAto 7F,, in particular; comparing the coolant amount control from the first cooling zone to
the fifth cooling zone when the casting speed was reduced from 0.8 m/min to 0.6 m/min and
34
:l:::j:i::1:'..,:--li;!.+i".¡:.!:-:.;-:--::-;..i;,.1:::::l=:ri;:_ir:r.:l:i,r:'.r_r.:,ll-:il:-:.-- r.-::. r.' r,,-: :':a-;!d.!.::: i!, -:-:::-:.:"-;::i¿=...î:;:Jü-;t:::::'ì
v/as then recovered from 0.6 m/min to 0.8 m/min, it was recognized that the coolant amount
from the first cooling zone to the fifth cooling zone was more appropriately adjustecl in
Exarnple I of FIGS. 6A to 6E than the comparative example of FIGS. TAto 1E such that the
difference between the slab surface temperature of the cooling zone exit and the target
temperature is recluced. As a result, according to the present invention, it was recognized that
the slab surface temperature could be controlled to the target temperature with high accuracy
even when the casting speed was changed.
l008rl
Example 2
In Example 2, the cooling control method according to the present invention was
applied to a case where the temperature target value of the third cooling zone was reduced by
20'C during casting. Note that the target temperature referred to a target value to be close to
the slab surface temperature predicted through the prediction step. In Exampl e 2, a
relationship between an actual measurement value of the slab surface temperature, the target
temperature, and time is illustratecl in FIG. 84, and a relationship befween the coolant amount
and time is illustrated in FIG. 88. In addition, a relationship between the casting speed and
time is illustrateclin FlG. BC.
As illustrated in FIGS. BA to 8C, when the temperature target value was reduced, and
the coolant amount of the third cooling zone increased, the slab surface temperature at the
third coolingzone exit gradually approached the target temperature obtained by reducing the
temperature by 20"C. ln contrast, reduction of the slab temperature at the fourth cooir¡e
zone entrance was compensated by lowering the temperature target value and then slightly
reducing the coolant amount of the fourth cooling zone. As a result, a change width of the
slab surface temperature at the fourth cooling zone exit was suppressed to 3oC. That is,
according to the present invention, it was recognized that the slab surface temperature could be
35
:!:j. r':::::i:::i: j: ::::::r1 jir:-:l¿:i.li ì';ì
controlled to the target temperature with high accuracy.
Note that, in Example 2, the coolant amount and the temperature were not changed in
the first or second cooling zone positioned in the upstrearn sicle of the thircl cooling zone in the
casting direction. For this reason, the results for the first and second cooling zones are not
illustrated, and only the results for the third and fourth cooling zones aÍe illustrated.
[0082]
Example 3
Assuming that the coohng was performed at a coolant amount set in advance through
calculation of the coolant amount, and the slab surface temperature at the fourth cooling zone
exit was expected to increase by 16"C from the target temperature, the coolant amount of the
fourth cooling zone was adjusted by sequentially estimating the actual heat transfer
coefficients using the cooling control method according to the present invention (in Example
3). In Example 3, a relationship between an actual measurement value of the slab surface
temperature, the target temperature, and time is illustrated in FIG. 9A., and a relationship
between the coolant amount and time is illustrated in FIG. 98. In addition, a relationship
between the casting speed and time is illustrated in FIG. 9C.
As illustrated in FIGS. 9A to 9C, the coolant amount for the fourth cooling zone was
controlled to increase relative to the initial setting value. As a result, it was possible to allow
the slab surface temperature at the fourth cooling zone exit to match the target value.
Accordingly, according to the present invention, it was recognized that the slab surface
temperature could be controlled to the target temperature with high accuracy.
Note that, in Example 3, the coolant amount and the temperature were not changed in
the first or second cooling zone positioned in the upstream side of the third cooling zone in the
casting direction. For this reason, the results for the first and second cooling zones are not
illustrated, and only the results for the third and fourth cooling zones are illustrated.
36
fReference Signs ancl Numerals]
[0083]
i mold
2 mist spray
3 flow-rate control valve
4 molten steel meniscus
5 slab
6 coolin g zoneboundary line (entrance or exit position)
7 slab surface temperature thermometer
8 casting speed rneasurement roll
9 continuous casting machine
10 cooling control device
10a tracking surface setting unit
10b slab target tempemture setting unit
10c temperature/solid-phase-rate estimation unit
10d heat transfer coefficient estimation unit
10e heat transfer/solidif,rcation model parameter correction unit
10f prediction suface getting unit
10g prediction unit
' . : 10h expected temperature influence coefücient prediction unit
10i slab surface reference temperature calculation unit
10j optimization problem coeffrcient matrix calculation unit
10k optimization problem solving unit
101 coolant amount change unit
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We claim:
1. A rnethod of controlling seconclary cooling of a continuous casting machine by clividing
a secondary cooling zone for cooling a slab extracted from a mold of the continuous casting
machine into a plurality of cooling zones in a casting direction ofthe slab and controlling a
coolant amount sprayed toward the slab in each cooling zone to contlol a slab surface
temperature, the method comprising:
a slab surface ternperature measurement step of measuring the slab surface temperature
at a predetermined temperature measurement point inside a strand during casting of the slab;
a casting speed recognition step of recognizing a casting speed of the continuous casting
machine;
a tracking surface setting step of setting a tracking surface as a target of calculation for
distributions of a temperature rnside a cross-sectional profile of the slab, the slab surface
temperature ; and asolid phase rate of the'slab at a predetermined interval from a position of a
molten metal surface inside the mold at least to a cooling zone exit of a secondary cooling
control target;
a slab target temperature setting step of setting a target value of the slab surface
temperature on the tracking surface;
a temperature/solicl-phase-rate estimation step of calculating and updating distributions
of the temperature inside the cross-sectional profile of the slab perpendicular to the casting
direction, the slab surface temperature, and the solid phase rate of the slab using a heat transfer
solidif,rcation:model based on a heat transfer equation whenever the fracking surface is shifted
by a predeterrnined interval in the casting direction of the slab as the casting progresses;
a heat transfer coefficient estimation step of calculating a heat transfer coefficient on a
slab surface used in the heat transfer solidification model by using a casting condition
including the coolant amount;
3B
a heat transfer/solidification model parameter correction step of corecting a pammeter
for the casting conclition in the heat transfer solidification model using a difference between
the slab surface temperature measured in the slab surface temperature measurement step and
the slab surface temperature estimated in the temperature/solid-phase-rate estimation step;
a prediction surface setting step of setting a prediction surface for predicting
distributions of the slab surface temperature at a subsequent timing, the temperature inside the
cross-sectional profile perpendicular to the casting direction, and the solid phase rate of the
slab at a predetermined constant interval in the casting direction out of a set of tr4cking
surfaces set in the tracking surface setting step;
a precliction step of, under assumption that the casting speed is not changed from the
current timing while an arbitrary prediction surface is shifted to a prediction surface position
neighboring to its downstream side from the current timrng as casting progresses, repeatedly
predrcting and updating distributions of the slab surface temperature, the temperature inside
¡file of the slab perpendicular to the casting direction, and the the cross-sectional profile castin¡ solid phase
rate of the slab when the prediction surface reaches a prediction surface position at the interval
used in the prediction surface setting step using the heat transfer solidification model;
an expected temperature influence coefticient prediction step of, under assumption that
the casting speed is not changed from the current timing whenever the arbitrary prediction
surface is shifted to a prediction surface position neighboring to its tlownstrearr side from the
cur¡ent timing as the casting progresses, predicting the slab surface temperature in each
tracking surface position in the middle of a path to the prediction surface position by assuming
that the coolant amount of each cooling zone is changed in a step function shape, obtaining a
deviation between the predicted slab surface temperature and the slab surface temperature
predicted in the prediction step, and 6þ1¿ining a change influence coefücient for the coolant
amount that changes in a step function shape using the deviation;
39
a slab surface reference temperature calculation step of calculating a reference target
temperaftrre detemiinecl clepenchng on time, the reference target temperature being between a
target value of the slab surface temperature set in the slab target temperature setting step and a
prediction value of the slab surface temperature at the timing when the prediction surface
predicted in the expected temperafure influence coefficient prediction step reaches the
prediction surface position;
an optim izafion problem coefficient matrix calculation step of setting the coolant
amount of each cooling z:one at the current timing as a determinant, calculating an expected
temperature influence coefficient in each prediction surface position where each prediction
surface of each of the prediction step and the expected temperature influence coefficient
prediction step passes, and a deviation between the reference target temperatures calculated in
the slab surfac reference temperature calculation step and the slab surface
.temperahrre
predicted in the prediction step, regarding an optimization problem for minimizing a sum of
the cleviations calculated for each prediction surface as a quadratic programming problem, and
calculating a coeffrcient matrix for the determinant in the quadratic programming problem;
an optimization problem solution step of numerically calculating the quadratic
programming problem to obtain an optimum value of a change amount of the coolant amount
that changes in a step function shape at the current timing; and
a coolant arnount change step of adding the optimum value to the coolant amount of a
current cooling zone to change the coolant amount,
wherein, by repeatedly changing the coolant amount in the coolant amount change step,
the slab surface temperature of the prediction surface in the prediction surface position is
controlled to the target value of the slab surface temperature set in the slab target temperature
setting step while each tracking surface at an arbitrary timing during the casting is shifted to a
cooling zone exit as the secondary cooling control target.
40
t:
a-
2. A device for controlling secondary cooling of a continuous casting machine by dividing
a secondary cooling zone for cooling a slab extracted from a molcl of the continuous casting
machine into a plurality of cooling zones in a casting direction of the slab and controlling a
coolant amount sprayed toward the slab in each cooling zone To control a slab surface
ternperature, the device comprising:
a slab surface temperahrre measurernent unit configured to measure the slab strrface
temperature at a predetennined temperature measurement point inside a strand during casting
of the slab;
a casting speed recognition unit configured to recognize a casting speed of the
continuous casting machine;
a tracking surface setting unit configured to set a tracking surface as a target of
calculation for distributions of a.temperature inside a cross-sectional profile of the slab; the
slab surface temperature , and asolid phase rate of the slab at a predetermined interval from a
position of a molten metal surface inside the mold at least to a cooling zone exit of a secondary
cooling control target;
a slab target temperature setting unit configured to set a target value of the slab surface
temperature on the tracking surface;
a temperature/solid-phase-rate estimation unit configured to calculate and update
distributions of the temperature inside the cross-sectional profile of the slab perpendicular to
the casting direction, the slab surface temperature, and the solid phase rate of the slab using a
heat transfer solidification model based on a heat transfer equation whenever the tracking
surface is shifted by a predetermined interval in the casting direction of the slab as the casting
progfesses;
a heat transfer coefficient estimation unit conf,rgured to calculate a heat transfer
4I
coefficient on a slab surface used in the heat transfer solidification model by using a casting
condition inclucling the coolant amount;
a heat transfer/solidification model pararneter correction unit configured to correct a
parameter for the casting condition in the heat transfer solidif,rcation model using a difference
between the slab surface temperahrre measured by the slab surface temperature measurement
unit and the slab surface temperature estimated by the temperature/solid-phase-rate estitnation
uniU
a prediction surface setting unit configured to set a prediction zurface for predicting
distributions of thå slab surface temperature at a subsequent timing, the temperature inside the
cross-sectional profile perpendicular to the casting direction, and the solid phase rate of the
slab at a predeterminecl constant interval in the casting direction out of a set of tracking
surfaces set by the tracking surface setting unit;
a prediction unit configured to; under assumption that the casting spepd is not changed
from the current timing while an arbitrary prediction surface is shiftecl to a prediction surface
position neighboring to its downsffeam side from the current timing as casting progresses,
repeatedly predict and update distributions of the slab surface temperature, the tempemture
inside the cross-sectional prof,rle of the slab perpendicular to the casting direction, and the
solid phase rate of the slab when the prediction surface reaches a prediction surface position at
the interval used by the prediction surface setting unit using the heat transfer solidification
model;
ap expected temperature influence coeff,rcient prediction unit configured to; under
assumption that the casting speed is not changed from the current timing whenever the
arbitrary prediction surface is shifted to a prediction surface position neighboring to its
downstueam side from the current timing as the casting progresses, predict the slab surface
temperature in each tracking surface position ín the middle of a path to the prediction surface
42
.i _¡, : t:r:r: :.1-:'_:-¡r:.::¡.:j:.1::
position by assuming that the coolant amount of each cooling zone is changed in a step
function shape, obtain a cleviation between the predicted slab suface temperature ancl the slab
surface temperature predicted by the prediction unit, and obtain a change influence coefflicient
for the coolant amount that changes in a step function shape using the deviation;
a slab surface reference temperature calculation unit configured to calculate a reference
target temperature detennined clepending on tirne, the reference target temperature being
between a target value of the slab surface temperature set by the slab target temperature setting
unit and a prediction value of the slab surface temperature at the timing when the prediction
surface predicted by the expected teÀperature influence coeffrcient precliction unit reaches the
prediction suface position;
an optimization problem coeff,rcient matrix calculation unit configurecl to set the coolant
amount of each cooling zone al the current timing as a determinant, calculate an expected
temperature influence coeffîcient in each prediction surface position where each prediction
surface of each of the prediction unit and the expected temperature influence coefficient
prediction unit passes, and a cleviation between the reference target temperatures calculated by
the slab surface reference temperature calculation unit and the slab surfâce temperature
predicted by the prediction unit, regarding an optirnization problem for minimizing a sum of
the deviations calculated for each prediction surface as a quadratic programming problem, and
calculating a coefficient matrix for the determinant in the quadratic programming problem;
an optimization problem solving unit configured to numerically calculate the quadratic
programming problem to
.obtain
an optimum value of a change amount of the coolant amount
that changes in a step function shape at the current timing; and
a coolant amount change unit configured to add the optimum value to the coolant
amount of a current cooling zone to change the coolant amount,
wherein, by repeatedly changing the coolant amount using the coolant amount change
:':.::l:-:-i:.::"-;:,--!:i.;;-¿.::::-I
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unit, the slab surface temperalure of the prediction surface in the prediction surface position is
controllecl to the target value of the slab surface temperature set by the slab target temperature
setting unit while each tracking surface at an arbitrary timing cluring the casting is shifted to a
cooling zone exit as the secondary cooling control target.