TECHNICAL FIELD
[0001] The present invention relates to an apparatus,
a method, and a program for detecting a molten metal
surface level in a continuous casting mold, and is
suitably used for detecting the molten metal surface
level in the continuous casting mold.
BACKGROUND ART
[0002] In operating a continuous casting facility,
it is necessary to detect the molten metal surface
level in the continuous casting mold and stably
control the molten metal surface level. This is
because the internal quality of a cast slab can be
improved by preventing overflow of molten steel and
roll-in of suspended substance. The molten metal
surface level refers to the height position of the
surface of the molten steel. As a technique of
detecting the molten metal surface level in the
continuous casting mold, there are techniques
described in Patent Literatures 1, 2. Note that the
continuous casting mold is abbreviated as a mold as
needed in the following description.
[0003] In Patent Literature 1, the following
technique is disclosed. A plurality of temperature
measurement elements are embedded in the mold at
- 1 -
regular intervals along the casting direction of the
mold (the height direction of the mold). The time
change rate of the temperature at a point of each
temperature measurement element is calculated to
detect a temperature measurement element (n)
exhibiting the maximum value of the time change rate.
A position exhibiting the maximum value of a quadric
curve linking the time change rate of the temperature
measurement element (n) and the time change rates of
two temperature measurement elements (n - 1), (n + 1)
adjacent to the temperature measurement element (n)
is obtained, and the position is regarded as the
molten metal surface level.
[0004] Besides, in Patent Literature 2, the
following technique is disclosed. A plurality of
thermocouples are embedded in the mold at intervals
along the casting direction of the mold (the height
direction of the mold). For detecting the molten
metal surface level, giving an initial temperature
distribution and deciding a temporary molten metal
surface level (division position) are performed first.
Upon decision of the temporary molten metal surface
level, the maximum heat flux and the minimum heat
flux at the temporary molten metal surface level are
calculated by analysis of heat conduction inverse
problems using the temperature changes measured by
the thermocouples. The maximum heat flux and the
minimum heat flux at the temporary molten metal
surface level are calculated with the temporary
- 2 -
molten metal surface level changed. Among the
calculated positions of the temporary molten metal
surface levels, the temporary molten metal surface
level where the difference between the maximum heat
flux and the minimum heat flux defined by performing
experiments in advance is smallest is regarded as the
actual molten metal surface level.
CITATION LIST
PATENT LITERATURE
[0005] Patent Literature 1: Japanese Laid-open
Patent Publication No. 53-26230
Patent Literature 2: Japanese Patent No. 4681127
SUMMARY OF INVENTION
TECHNICAL PROBLEM
[0006] However, the technique described in Patent
Literature 1 is based on the empirical rule that the
position where the temperature in the casting
direction of the mold is maximum exists near the
molten metal surface, and this position is in a
certain correlation with the molten metal surface
level. In the case based on the empirical rule as
described above, the detection accuracy of the molten
metal surface level may decrease.
[0007] Besides, in the technique described in Patent
Literature 2, the initial value (initial condition)
of the temperature distribution is necessary when
performing the analysis of heat conduction inverse
problems by the non-stationary heat conduction
equation. F11rther, the heat flux in the casting
- 3 -
---~-----
direction is calculated as a discrete value.
Accordingly, also in the technique described in
Patent Literature 7, the detection accuracy of the
molten metal surface level may decrease. In
particular, when the state of the molten steel inside
the mold rapidly changes to increase the temporal
change of the heat flux, the detection accuracy of
the molten metal surface level may decrease.
[0008] The present invention has been made in
consideration·of the above poir1ts, and its objeot is
to increase the detection accuracy of the molten
metal surface level in the continuous casting mold.
SOLUTION TO PROBLEM
[0009] An apparatus for detecting a molten metal
surface level in a continuous casting mold of the
present invention includes: a temperature acquisition
means which acquires temperatures measured by a
plurality of temperature measurement means embedde'd
in the continuous casting mold along a casting
direction of the continuous casting mold; a heat flux
derivation means which derives a value of a casting
direction component of a heat flux vector on an inner
wall surface of the continuous casting mold, based on
a result of performing analysis of non-stationary
heat conduction inverse problems from the
temperatures derived by the temperature acquisition
means; and a molten metal surface level derivation
means which derives a molten metal surface level
inside the continuous casting mold, based on the
- 4 -
value of the casting direction component of the heat
flux vector on the inner wall surface of the
continuous casting mold derived by the heat flux
derivation means, wherein the molten metal surface
level derivation means derives, as the molten metal
surface level, a position where an absolute value of
the value of the casting direction component of the
heat flux vector whose vector of the casting
direction component is in a direction opposite to the
casting direction is maximum.
[0010] A method for detecting a molten metal surface
level in a continuous casting mold of the present
invention includes: a temperature acquisition step of
acquiring temperatures measured by a plurality of
temperature measurement steps embedded in the
continuous casting mold along a casting direction of
the continuous casting mold; a heat flux derivation
step of deriving a value of a casting direction
component of a heat flux vector on an inner wall
surface of the continuous casting mold, based on a
result of performing analysis of non-stationary heat
conduction inverse problems from the temperatures
derived by the temperature acquisition step; and a
molten metal surface level derivation step of
deriving a molten metal surface level inside the
continuous casting mold, based on the value of the
casting direction component of the heat flux vector
on the inner wall surface of the continuous casting
mold derived by the heat flux derivation step,
- 5 -
wherein the molten metal surface level derivation
step derives, as the molten metal surface level, a
position where an absolute value of the value of the
casting direction component of the heat flux vector
whose vector of the casting direction component is in
a direction opposite to the casting direction is
maximum.
[0011] A program of the present invention causes a
computer to execute: a temperature acquisition step
of acquiring temperatures measured by a plu~ality of
temperature measurement steps embedded in a
continuous casting mold along a casting direction of
the continuous casting mold; a heat flux derivation
step of deriving a value of a casting direction
component of a heat flux vector on an inner wall
surface of the continuous casting mold, based on a
result of performing analysis of non-stationary heat
conduction inverse problems from the teiliperatures
derived by the temperature acquisition step; and a
molten metal surface level derivation step of
deriving a molten metal surface level inside the
continuous casting mold, based on the value of the
casting direction component of the heat flux vector
on the inner wall surface of the continuous casting
mold derived by the heat flux derivation step,
wherein the molten metal surface level derivation
step derives, as the molten metal surface level, a
position where an absolute value of the value of the
casting direction component of the heat flux vector
- 6 -
whose vector of the casting direction component is in
a direction opposite to the casting direction is
maximum.
ADVANTAGEOUS EFFECTS OF INVENTION
[0012] According to the present invention, it is
possible to increase the detection accuracy of the
molten metal surface level in the continuous casting
mold.
BRIEF DESCRIPTION OF DRAWINGS
[0013] [Fig. 1] Fig. 1 is a v.iew illustrating an
example of a configuration of a system for detecting
a molten metal surface level in a continuous casting
mold.
[Fig. 2] Fig. 2 is a diagram illustrating an
example of a functional configuration of the
apparatus for detecting the molten metal surface
level in the continuous casting mold.
[Fig. 3A] Fig. 3A is a chart illustrating an
example of a two-dimensional cross section of a space
x-time t of a coordinate system of non-stationary
heat conduction inverse problems.
[Fig. 3B] Fig. 3B is a chart illustrating an
example of a two-dimensional cross section of a space
x-space y of the coordinate system of the nonstationary
heat conduction inverse problems.
[Fig. 4] Fig. 4 is a flowchart for explaining an
example of the operation of the molten metal surface
level detection apparatus.
[Fig. 5] Fig. 5 is a diagram illustrating an
- 7 -
·-----~-.. ·---------
example of a hardware configuration of the molten
metal surface level detection apparatus.
[Fig. 6] Fig. 6 is a view illustrating positions
of thermocouples in an example.
[E'ig. 7A] Fig. 7A is a chart conceptually
illustrating an example of the relation between a
value of a y-axis direction component of a heat flux
vector on an inner wall surface of the mold and a
position in a y-axis direction.
[Fig. 7B] Fig. 7B is a chart conceptually
illustrating the relation between a temperature in
the mold and the position in the y-axis direction.
[Fig. 8j Fig. 8 is a view illustrating an example
of a configuration of an apparatus for actually
measuring the molten metal surface level.
[Fig. 9] Fig. 9 is a view illustrating a molten
metal surface level detected in an inventive example,
a molten metal surface level detected by an existing
method, and an actually measured molten metal surface
level.
DESCRIPTION OF EMBODIMENTS
[0014] (Molten metal surface level detection system
in a continuous casting mold)
Fig. 1 is a view illustrating an example of a
configuration of a system for detecting a molten
metal surface level in a continuous casting mold.
Fig. 1 illustrates a cross section of a continuous
casting machine obtained by cutting it along its
height direction (a y-axis direction)
- 8 -
~~--
In Fig. 1, the continuous casting machine has a
tundish 11, an immersion nozzle 12, a mold 13, and
pinch rolls 14a to 14d: Note that the continuous
casting machine can be realized by a publicly-known
technique. Accordingly, the detailed description of
the continuous casting machine will be omitted here.
[0015) The tundish 11 temporarily stores molten
metal M supplied from a ladle.
The mold 13 is disposed below the tundish 11 with
a gap intervening between the mold 13 and the tundish
11. The mold 13 has, for example, two short side
parts 13a, 13b, and two long side parts. The two
short side parts 13a, 13b are arranged with a gap
intervening between them to be opposed to each other
in a width direction (an x-axis direction). The two
long side parts are arranged with a gap intervening
between them to be opposed to each other in a depth
direction (a direction perpendicular to an x-axis and
a y-axis). A region surrounded by the two long side
parts and the two short side parts 13a, 13b becomes a
region in a hollow rectangular parallelepiped shape.
This region becomes a region inside the mold 13.
Further, a groove is formed on an outer wall surface
of the mold 13. The mold 13 is water-cooled by
running water through the groove. Note that only the
short side parts of the long side parts and the short
side parts are illustrated in Fig. 1 for convenience
of illustration.
[0016) The immersion nozzle 12 injects the molten
- 9 -
metal M stored in the tundish 11 to the inside of the
mold 13. The immersion nozzle 12 is disposed such
that its base end is located at the bottom surface of
the tundish 11 and a predetermined region on the tip
end side is located inside the mold 13. Further, the
inside of the immersion nozzle 12 and the inside of
the tundish 11 arc communicated with each other.
Note that the supply amount of t~e molten metal M to
be supplied from the tundish 11 to the immersion
nozzle 12 is adjusted by a sliding nozzle or a
stopper.
[0017] A plurality of pairs of pinch roll~ 14a to
14d are arranged along a conveyance path for steel
drawn downward from the mold 13. Note that only two
pairs of pinch rolls 14a to 14d are illustrated in
Fig. 1. However, more pinch rolls are actually
arranged according to the length of the conveyance
path. Outside the pinch rolls l4a to 14d, a
plurality of cooling sprays are arranged. The
plurality of cooling sprays spray cooling water for
cooling the steel drawn downward from the mold 13, to
the steel.
[0018] As described above, the molten metal injected
to the inside of the mold 13 is cooled by the mold 13
and solidified due to formation of solidified shells
15a, 15b from its surface. The steel having a
surface being the solidified shel~s 15a, 15b and an
inside being not solidified is continuously drawn
from a lower end portion of the mold 13 while being
- 10 -
I
sandwiched between the pinch rolls 14a to 14d. In
the process of being drawn from the mold 13 in this
manner, the cooling water sprayed from the cooling
sprays proceeds cooling of the steel to solidify the
steel up to the inside. The thus-solidified steel is
cut into a predetermined size on the dbwnstream side
of the continuous casting machine, whereby a cast
slab different in shape of the cioss section such as
slab, bloom, billet or the like is manufactured.
[0019] In manufacturing the cast slab by the.
continuous casting machine as described above, powder
17 is added as needed to the molten metal inside the
mold 13. A thin film of the powder 17 exists also
between an inner wall surface of the mold 13 and the
solidified shells 15a, 15b as well as on the surface
of the molten metal inside the mold 13. The addition
of the powder 17 in this manner achieves retention of
heat of the molten metal, prevention of oxidation of
the molten metal, absorption of inclusion in the
molten metal, securement of lubricity of the
solidified shells 15a, 15b, and adjustment of removal
of heat of the molten metal. Uniform generation of
the solidified shells 15a, lSb near the meniscus in
the mold 13 in the above manner prevents surface
crack of the solidified shells lSa, 15b and prevents
seizing between the mold 13 and the solidified shells
lSa, 15b.
[0020] In the mold 13, a plurality of thermocouples
18 are embedded along the casting direction (the y-
- 11 -
<>xis direction). The number of the plurality of
thermocouples 18 is preferably three or more.
According to the calculation accuracy of a laterdescribed
heat flux, the number of the plurality of
thermocouples 18 and the interval between two
adjacent thermocouples 18 can be decided. Further,
in the example illustrated in Fig. 1, the plurality
of thermocouples 18 are embedd€d in a region
relatively closer to the inner wall surface of the
inner wall surface and the outer wall surface of the
mold 13. However, the plurality of thermocouples 18
do not always need to be embedded in such a region as
long as they are embedded in the mold 13. As
illustrated in Fig. 1, a case wh€re the plurality of
thermocouples 18 are embedded in the short side part
13a will be described as an example in this
embodiment. However, a plurality of thermocouples
may be embedded in at least any one of the short side
part 13b and the two long side parts in addition to
or instead of the short side part 13a. The inner
wall surface of the mold 13 is called an operation
surface, and the outer wall surface thereof is called
a water cooled surface. The surface in contact with
the molten metal of the surfaces of the mold 13 is
the operation surface. However, in the case where
the powder 17 is added as illustrated in Fig. 1, the
surface in contact with the powder 17 of the surfaces
of the mold 13 is the operation surface.
(0021] (Apparatus 200 for detecting a molten metal
- 12 -
surface level in a continuous casting mold)
Fig. 2 is a diagram illustrating an example of a
functional configuration of the apparatus 200 for
detecting the molten metal surface level in the
continuous casting mold. The apparatus for detecting
the molten metal surface level in the continuous
casting mold is abbreviated as a molten metal surface
level detection apparatus as needed.
The molten metal surface level detection
apparatus 200 performs analysis of non-stationary
heat conduction inverse problems using the
temperatures measured by the plurality of
thermocouples 18. The non-stationary heat conduction
inverse problems here refer to a problem of
estimating a boundary condition or an initial
condition such as the temperature and the heat flux
in a boundary region with the temperature information
inside the region having been known, based on the
non-stationary heat conduction equation dominating a
calculation area. In contrast to this, nonstationary
heat conduction forward problems refer to
a problem of estimating the temperature information
inside the region, based on a known boundary
condition.
[0022] The molten metal surface level detection
apparatus 200 calculates the value of a component in
the y-axis direction (the casting direction of the
mold 13) of a heat flux vector on the inner wall
surface of the mold 13 using an
- 13 -
interpolation/extrapolation temperature function
obtained by performing the analysis of non-stationary
heat conduction inverse problems. As will be
described later, the interpolation/extrapolation
temperature function is a function indicating the
temperature of the mold 13 at a position (x, y) and a
time t.
[0023] The molten metal surface level detection
apparatus 200 detects a molten metal surface level,
based on the ~alue of a y-axis direction component of
the heat flux vector on the inner wall surface of the
mold 13. The molten metal surface level is the
height position (a position in the y-axis direction)
of the surface of the molten metal irrside the mold 13.
[0024] The role of the mold 13 is cooling and
solidification of the molten metal. Therefore, in
discussing the detection of the molten metal surface
level by performing the analysis of non-stationary
heat conduction inverse problems, the behavior of the
heat flux in an x-axis direction (a heat removal
direction of the mold 13) has attracted attention,
whereas the behavior of the heat flux in the y-axis
direction (the casting direction of the mold 13) has
not attracted attention. Further, the value of the
y-~xis direction component of the heat flux vector is
smaller than the value of an x-axis direction
component. Therefore, by the method of deriving the
heat flux taking a discrete value as in the technique
described in Patent Literature 2, if the value of the
- 14 -
y-axis direction component of the heat flux vector is
used, the error increases, causing a further decrease
in the calculation accuracy of the heat flux. From
the above, in the case of deriving the heat flux of
the mold 13 by performing the analysis of nonstationary
heat conduction inverse problems including
the case of detecting the molten metal surface level
by performing the analysis of non~stationary heat
conduction inverse problems, the value of the x-axis
direction component of the heat flux vector ha& been
used heretofore.
[0025] In contrast to the above, the present
inventors have reached an idea of detecting the
molten metal surface level, based on the inference
that "on the molten metal surface inside the mold 13,
the magnitude of the vector in a direction opposite
(namely, facing the normal direction to the molten
metal surface) to the casting direction of the vector
of the y-axis direction component of the heat flux
vector becomes large as compared with that at the
other portion of the mold 13 due to the influence of
the heat removal by the powder 17" because the powder
17 is supplied onto the molten metal surface inside
the mold 13. Under such an idea, the molten metal
surface level detection apparatus 200 in this
embodiment has been realized. Hereinafter, an
example of a concrete configuration of the molten
metal surface level detection apparatus 200 in this
embodiment will be described.
- 15 -
[0026] The molten metal surface level detection
apparatus 200 has a temperature acquisition unit 201,
a heat flux derivation unit 202, a heat flux
derivation unit 20~, and a molten metal surface level
derivation unit 203.

The temperature acquisition unit ~01 receives
input of temperatures [K] measured by the plurality
of thermocouples 18, and outputs the temperatures
measured at the same time by the plurality of
thermocouples 18. The temperature acquisition unit
201 performs such output of the temperatures at each
predetermined sampling time. For example, the
temperature acquisition unit 201 receives input of
and outputs the temperatures measured by the
plurality of thermocouples 18 every time the sampling
time elapses.
[0027]
Based on the temperatures outputted from the
temperature acquisition unit 201, an
interpolation/extrapolation temperature function uA(x,
y, t) for estimating the temperature of the mold 13
is used to make a mathematical expression of
predicting the temporal change in temperature
distribution on a two-dimensional cross section in
the casting direction (the y-axis direction)-the heat
removal direction (the x-axis direction) of the mold
13.
[0028] Fig. 3A is a chart ill11strating an example of
- 16 -
a coordinate system of the non-stationary heat
conduction inverse problems. Fig. 3A illustrates a
definition point of the information amount on the
two-dimensional cross section of a space x-time t at
a certain position in the y-axis direction. Fig. 3B
is also a chart illustrating an example of the
coordinate system of the non-stationary heat
conduction inverse problems. Fig. 3B illustrates a
definition point of the information amount on the
two-dimensional cross section of a space x-space y at
a certain time t. Fig. 3A and Fig. 3B illustrate the
two-dimensional cross sections of the same threedimensional
coordinates (coordinates of a space xspace
y-time t).
[0029] In Fig. 3A and Fig. 3B, the x-axis is an axis
where the inner wall surface of the mold 13 is x = 0,
and indicates the position in the heat removal
direction of the mold 13. The y-axis is an axis
where the upper end of the mold 13 is y = 0, and
indicates the position in the casting direction of
the mold 13. The x-axis and the y-axis are space
axes. The t-axis is a time axis.
[0030] In Fig. 3A and Fig. 3b, plots indicated by
black circles are definition points of the
information amounts, respectively. The definition
point of the information amount indicates the
position of the thermocouple 18 and the time when the
temperature was measured by the thermocouple 18. The
information amount at the definition point includes
- 17 -
the temperature measured by the thermocouple 18.
[0031] Plots indicated by broken lines are also
definition points of the information amounts,
respectively. The definition point of the
information amount indicates the position on the
outer wall surface of the mold 13 and the time when
the heat flux on. the outer wall surface is estimated.
In this embodiment, a case where ~ temperature
measurement means such as the thermocouple is not
provided on the outer wall surface of the mold 13
will be described as an example. Hence, the
information amount at the definition point is
regarded as the heat flux decided with a heat
transfer coefficient y between the material
constituting the mold 13 and water and a water
temperature Uw having been known.
[0032] The plots indicated by the black circles and
the plots indicated by the broken lines in the above
are used as the definition points of the information
amounts. More specifically, each of the points on
the three-dimensional coordinates of the x-axis-the
y-axis-the t-axis represented by the plots indicated
by the black circles and the plots indicated by the
broken lines illustrated in Fig. 3A and the plots
indicated by the black circles and the plots
indicated by the broken lines illustrated in Fig. 38
is the definit~on point of the information amount.
[0033] In Fig. 3A, a timing tN is a timing when the
latest temperatures were measured by the plurality of
- 18 -
thermocouples 18. In Fig. 3A, a case where every
time the temperatures measured by the plurality of
thermocouples 18 are acquired, seven temperature
measurement timings (seven timings such as timings to
to t.l are employed in sequence from the new one as a
time t when the definition point of the information
amount is decided will be described as an example.
More specifically, when the temperatures measured by
the plurality of thermocouples 18 are newly acquired,
the heat flux derivation unit 202 excludes the
definition point of the information amount including
the oldest temperature measurement timing of the
seven temperature measurement timings, from the seven
definition points of the information amounts. The
heat flux derivation unit 202 then adds the
definition point of the information amount including
the latest temperature measurement timing to the
seven definition points of the information amounts.
Note that the number of times t deciding the
definition points of the information amounts is not
limited to seven.
[0034] Besides, in Fig. 3B, a case where seven
thermocouples 18 are arranged at regular intervals
along the y-axis direction as the plurality of
thermocouples 18 will be illustrated as an example.
However, the interval between two thermocouples 18
adjacent to each other does not need to be the
regular interval. Besides, the number of the
plurality of thermocouples 18 is not limited to seven.
- 19 -
[0035] The heat flux derivation unit 202 derives a
weight vector Aj included in the
.interpolation/extrapolation temperature fu.nction uA (x,
y, t) on the basis of the above definition points of
the information amounts.
I
An example of the interpolation/extrapolation
temperature function uA(x, y, t) will be described
here.
First, the quadratic non-stationary heat
conducti.on equation is expressed by the following (1)
expression.
[0036] [Expression 1]
O (x-xi, y-yi, t-ti) is a basis function
decided by the following (5) expression and (6)
expression.
[0049] [Expression 5]
¢ (x, y, t) = f(x, Y. t+T) . . . ( 5)
( 6)
[0050] In the (6) expression, H (t) is a Heaviside
function. The ( 6) expression is an expression
expressed in the form of a fundamental solution
satisfying the quadratic non-stationary heat
conduction equation expressed in the (1) expression.
Note that the fundamental solution is a solution (the
temperature u) of the quadratic non-stationary heat
conduction equation when the initial condition of the
temperature u is expressed by a o function. In the
(5) expression, T is a parameter for adjusting the
diffusion profile of the fundamental solution of the
quadratic non-stationary heat conduction equation,
and is set in advance. T is a value more than 0.
[0051] As described above, the basis function (x-xi•
- 25 -
y-yi, t-ti) is a function expressed in ·the form of the
fundamenta~ solution satisfying the quadratic nonstationary
heat conduction equation on the basis of
the center point j (the reference vector (Xj, Yi) and
the reference time ti).
[0052] ~j is a weight vector representing tho weight
of the basis function ¢ (x-xi, y-yi, t-ti) with respect
to the interpolation/extrapolation temperature
function uA(x, y, t). The weight vector ~i is decided
by the balance between the influence of the basis
function ¢ (x-xj, y-yj, t-ti) on the
interpolation/extrapolation temperature function uA(x,
y, t) and the influence of another basis function
¢(x-xi, y-yi, t-ti) different from the above basis
function ¢ (x-xi, y-yi, t-ti) on the
interpolation/extrapolation temperature function uA(x,
y' t) . The basis function ¢ (x- Xj, y-yi, t-tj) exists
for each center point j, and the weight vector ~i also
exists for each center point j.
[0053] As described above, the
interpolation/extrapolation temperature function uA(x,
y, t) is expressed by the total sum of the values at
respective center points j of the products of the
basis function ¢ (x- xi, y-yi, t-ti) and the weight
vector ~i.
The weight vector ~i is expressed by the
following (7) expression to (10) expression.
[0054] [Expression 6]
- 26 -
A A.= b (7)
A=
a¢ f3 ax (xk-Xj, Yk-Yj, tk-tj) + Y¢ (~-Xj, Yk-Yj. tk-tj)
¢ (X8 -Xj, Ys-Yj• t 8-tj)
(8)
( 9)
(10)
[0055] In the (8) expression and the (10) expression,
k is a variable that identifies the definition point
of the information amount, and is an integer from 1
tom(k=1, ... ,m). s is a variable that
identifies the definition point of the information
amount, and is an integer from m + 1 to m + 1 (s m
+ 1, . , m + 1) j is an integer from 1 to m + 1 (j
1' . ' m + 1)
A matrix A is a (m + 1) x (m + l) matrix. b and
A are (m + l) -dimensional column vectors. As
described above, (m + 1) is the number of center
points j.
[0056] In the (8) expression, "j33¢/3x (xk- Xj, Yt-Yj,
represents a k-row and j-column component of the
matrix A, and "cj:> (xs- Xj, Ys-yj, ts- tj)" represents an
s-row and j-column component of the matrix A.
- 27 -
[0057] To g, in [] of b [], g (t) expressed in the
(2) expression is given. g, in [] represents a k-row
component of a b. To hs m in [ l of b = [],
h(t) expressed in the (3) expression is given.
in [] represents an s-row component of the matrix b.
[0058] As described above, k is a variable that
identifies the definition point of the information
amount, and is an integer from 1 to m (k = 1, .. ' m)
m is expressed by np 1 x nt. np1 is the number of the
center points j on the outer wall surface of the mold
1.3. The coordinate on the x-axis is decided so that
the coordinate on the x-axis on the inner wall
surface of the mold 13 is ''0'' and the coordinate on
the x-axis on the outer wall surface thereof is "1".
Accordingly, in the (8) expression, x• becomes "1".
[0059] The (7) expression to the (10) expression are
expressions for deriving the weight vector Ai by
substituting the information on the definition point
of the information amount into simultaneous equations
of the (2) expression and the (4) expression and
solving the simultaneous equations so as to satisfy
the quadratic non-stationary heat conduction equation
of the (1) expression, the boundary condition on the
outer wall surface of the mold 13 of the (2)
expression, the thermocouple temperature function
(the temperature measured by the thermocouple in the
mold 13 at each position (x*, y*) and at each time t)
of the (3) expression, and the
interpolation/extrapolation temperature function of
- 28 -
---·---~~----·--------~
the (4) expression. The information on the
definition point of the information amount
substituted into the simultaneous equations includes
the position of the definition point of the
information amount, the temperature by the
thermocouple 18, the temperature measurement timing
of the thermocouple 18, the water temperature u., the
heat conductivity ~ of the material constituting the
mold 13, the heat transfer coefficient y between the
material constituting the mold 13 and water, and the
thermal diffusion coefficient a of the material
constituting the mold 13. The water temperature u.,
the heat conductivity ~ of the material constituting
the mold 13, the heat transfer coefficient y between
the material constituting the mold 13 and water, and
the thermal diffusion coefficient a of the material
constituting the mold 13 may be made different
depending on the definition point of the information
amount, or may be made the same. Besides, in solving
the simultaneous equations of the (2) expression and
the (4) expression, the position of the center point
j is also substituted into the simultaneous equations.
[0060] The heat flux derivation unit 202 derives the
weight vector Aj by the (7) expression to the (10)
expression in the above manner.
The heat flux derivation unit 202 performs the
above processing every time of acquiring the
temperature from the temperature acquisition unit 201.
In this embodiment, the vaJue qy of they-axis
- 29 -
direction component of the heat flux vector is
expressed by the following (.11) expression.
[0061] [Expression 7]
_
13
.auA(x=O,y,t)
qy- ay
m+l y-yj
= -/3 L A·--::---t=====H(t-t-)
j=l J4a3 (t-tjHn(t-'-tj) J
(11)
[0062] Accordingly, the heat flux derivation unit
202 derives the value qy of the y-axis diroctiori
component of the heat flux vector on the innei wall
surface of the mold 13 by substituting the heat
conductivity ~ of the material constituting the mold
13, the thermal diffusion coefficient a of the
material constituting the mold 13, the reference time
tj, the number m + l of the center points j, and the
weight vector hj derived as described above, into the
(11) expression.
[0063]
The molten metal surface level derivation unit
203 derives the relation between the value qy of the
y-axis direction component of the heat flux vector
and the position in the y-axis direction, from the
value qy of the y-axis direction component of the heat
flux vector derived by the heat flux derivation unit
202. The molten metal surface level derivation unit
203 derives, from the relation, the position where
the value qy of the y-axis direction component of the
- 30 -
heat flux vector has a negative value and its
absolute value is maximum (namely, minimum), as the
molten metal surface level. In this embodiment, the
y-axis is defined as illustrated in Fig. 1.
Accordingly, the position where the value qy of the yaxis
direction component of the heat flux vector on
the inner wall surface of the mold 13 becomes minimum
(the absolute value of negative ~alues is maximum) is
the molten metal surface level. Note that when the
y-axis is defined to be the direction opposite-to the
directiori illustrated in Fig. 1, the position where
the value qy of the y-axis direction component of the
heat flux vector on the inner wall surface of the
mold 13 is maximum is the molten metal surface level.
As described above, the molten metal surface level
derivation unit 203 derives, as the molten metal
surface level, the position where the absolute value
of the value qy of the y-axis direction component of
the heat flux vector whose y-axis component vector is
in the direction opposite to the casting direction
(namely, directed in the normal direction to the
molten metal surface) is maximum.
[0064]
The output unit 204 outputs the information on
the molten metal surface level derived by the molten
metal surface level derivation unit 203. As the
output form of the information on the molten metal
surface level, at least one of display on a computer
display, storage into a storage medium in the molten
- 31 -
metal surface level detection apparatus 200 or a
portable storage medium, and transmi~sion tb an
external device can be employed.
[0065] (Flowchart)
Next, an example of the operation of the molten
metal surface level detection apparatus 200 in this
embodiment will be described referring to the
flowchart in Fig. 4.
At Step S401, the temperature acquisition unit
201 acquires the temperatures measured by the
plurality of thermocouples 18.
[0066] Next, at Step S402, the heat ·flux derivation
unit 202 determines whether the required number of
temperatures for deriving the weight vector Aj have
been acquired or not. Concretely, the heat flux
derivation unit 202 waits until one temperature as
the number of the definition point of the information
amount with respect to the thermocouple 18 is
acquired. In the examples illustrated in Fig. 3A and
Fig. 3B, the heat flux derivation unit 202 waits
until 49 temperatures are acquired because there are
seven definition points of the information amounts in
the y-axis direction and there are seven definition
points of the information amounts in the t-axis
direction. Note that in the case where the 49
temperatures have been already acquired, when the
temperatures corresponding to the seven definition
points of the information amounts in the y-axis
direction are acquired at the same time, the heat
- 32 -
flux derivation unit 202 deletes the temperatures at
the oldest time among the t~mperatures corresponding
to the seven definition points of the information
amounts in the y-axis direction at the same time, and
adds the temperatures acquired this time.
[0067] When the required number of temperatures for
deriving the weight vector Aj have not been acqujred
as a result of the determination, the flow returns to
Step S401. Then, the processing at Steps S401 and
S402 is repeatedly performed until the required
number of temperatures for deriving the weight vector
Aj are acquired. When the required nUmber of
temperatures for deriving the weight vector Aj are
acquired, the flow proceeds to Step S403.
[0068] When proceeding to Step S403, the heat flux
derivation unit 202 derives the weight vector Aj by
the (7) expression to the (10) expression.
Next, at Step S404, the heat flux derivation unit
202 derives the value qy of the y-axis direction
component of the heat flux vector on the inner wall
surface of the mold 13 by the (11) expression.
[0069] Next, at Step S405, the molten metal surface
level derivation unit 203 derives the relation
between the value qy of the y-axis direction component
of the heat flux vector and the position in the yaxis
direction. The molten metal surface level
derivation unit 203 derives, from the derived
relation, the position where the value qy of the yaxis
direction component of the heat flux vector has
- 33 -
a negative value and its absolute value is maximum
(namely, minimum), as the molten metal surface level.
[0070] Next, at Step S406, the output unit 204
outputs the information on the molten metal surface
level derived by the molten metal surface level
derivation unit 203.
Next, at Step S407, the molten metal surface
level detection apparatus 200 determines whether to
end the derivation of the molten metal surface level.
This determination is performed,' for example, based
on the operation by an operato~ to t.he molten metal
surface level detection apparatus 200.
[0071] When the derivation of the molten metal
surface level is not ended as a result of the
determination, the flow returns to Step S401. Then,
the processing at Steps S401 to S407 is repeatedly
performed every time temperatures are newly acquired
at Step S401.
On the other hand, when the derivation of the
molten metal surface level is ended, the processing
by the flowchart in Fig. 4 is ended.
[0072] (Hardware of the apparatus 200 for detecting
the molten metal surface level in the continuous
casting mold)
Fig. 5 is a diagram illustrating an example of a
hardware configuration of the molten metal surface
level detection apparatus 200.
As illustrated in Fig. 5, the molten metal
surface level detection apparatus 200 has a CPU
- 34 -
(Central Processing Unit) 501, a ROM (Read Only
Memory) 502, a RAM (Random Access Memory) 503, a PD
(Pointing Device) 504, ·an HD (Hard Disk) 505, a
display device 506, a speaker 507, an I/F (Interface)
508, and a system bus 509.
[0073] The CPU 501 centrally controls the operation
in the molten metal surface level detection apparatus
200. The CPU 501 controls the c6mponents (502 to
508) of the molten metal surface level detection
apparatus 200 via the system bus 509.
The ROM 502 stores a BIOS (Bas.ic Input/Output
System) and an operating system program (OS) being
control programs of the CPU 501, and programs
required for the CPU 501 to execute the processing by
the above-described flowchart illustrated in Fig. 4
and so on.
[0074] The RAM 503 functions as a main memory, a
work area and so on of the CPU 501. For executing
the processing, the CPU 501 realizes various
operations by loading necessary computer programs,
information and so on from the ROM 502 and the HD 505
into the RAM 503 and executing processing on the
computer programs, the information and so on. The
computer program of executing the processing in the
above-described flowchart in Fig. 4 may be stored in
the HD 505.
The PD 504 is composed of, for example, a mouse,
a keyboard or the like, and constitutes an operation
input means for the operator to perform an operation
- 35 -
input to the molten metal surface level detection
apparatus 200 as needed.
The HD 505 constitutes a storage means that
stored various kinds of information, data, files and
so on.
The display device 506 constitutes a display
means that displays various kinds of information and
images, based on the control of the CPU 501.
The speaker 507 constitutes a sound output means
that outputs sound relating to various kinds tif
information, based on the control o{ the CPU 501.
[0075] The I/F 508 performs communication of varioris
kinds of information and so on with the external
device, based on the control of the CPU 501. The
temperature measured by the thermocouple 18 is
inputted into the molten metal surface level
detection apparatus 200 via the I/F 508.
The system bus 509 is a bus for connecting the
CPU 501, the ROM 502, the RAM 503, the PD 504, the HD
505, the display device 506, the speaker 507, and the
I/F 508 so that they can communicate with one another.
[0076] (Examples)
The molten metal surface level detected by the
method of this embodiment, the molten metal surface
level detected by an existing method, and the
actually measured molten metal surface level were
compared. As illustrated in Fig. 6, the plurality of
thermocouples 18 are embedded in the short side part
13a of the mold 13. As illustraled in Fig. 6 1 the
- 36 -
plurality of thermocouples 18 do not have to be
embedded in the mold 13 accurately along the y-axis
direction. However, the above-described weight
vector Aj is derived with the coordinates on the xaxis
of the thermocouples 18 set to the same value.
More specifically, the positions in the x-axis
direction of the thermocouples 18 do not have to be
precis~ly the same as long as the~ do not affect the
accuracy of the weight vector Aj. Further, the water
temperature on the in-side (upper side) of the mold
13 and the water temperature on the out-side (lower
sidS) of the mold 13 were measured and their average
value was calculated and regarded as the temperature
of the cooling water.
[ 0077] In the method of this embodiment, the value qy
of the y-axis direction component of the heat flux
vector on the inner wall surface of the mold 13 is
derived as described above. Then, the position where
the absolute value of the value qy of the y-axis
direction component of the heat flux vector whose yaxis
component vector is in the direction opposite to
the casting direction is maximum is determined as the
molten metal surface level L. Fig. 7A conceptually
illustrates an example of the relation, obtained by
the method of this embodiment, between the value qy of
the y-axis direction component of the heat flux
vector on the inner wall surface of the mold 13 and
the position in the y-axis direction.
[0078] On the othBr hand, in the existing method,
- 37 -
the temperature distribution in the mold 13 is
calculated, and the position of a maximum temperature
(T~axl • 0.65 i.s determined as the molten metal
surface level L, based on the empirical rule. Fig.
7B conceptually illustrates an example of the
relation, obLairted by the existing method, between
the temperature in the mold 13 and the position in
the y-axis direction.
[0079] The molten metal surface level was actually
measured using an apparatus illustrated in Fig. 8. A
float 801 is floated on the molten met~l surface ~f
the molten steel inside the mold, and a rod 802 is
disposed on the float 801. ~·urther, an oscillation
measuring jig 803 is disposed. Then, the movement of
the tip of the rod 802 and the movement of the tip of
the oscillation measuring jig 803 are image-captured
by a video camera 804. Image processing is performed
on the image captured by the video camera 804 to
digitalize and record the displacement in the y-axis
direction of the molten metal surface. From the
displacement in the y-axis direction of the molten
metal surface, the molten metal surface level was
obtained.
[0080] Fig. 9 illustrates the molten metal surface
level detected by the method of this embodiment, the
molten metal surface level detected by the existing
method, and the actually measured molten metal
surface level. The horizontal axis indicates time
and the longitudinal axis indicates the molten metal
- 38 -
surface level.
The existing method extremely decreases in
detection accuracy when the actually measured molten
metal surface level rises, and cannot follow the
actually measured value.
In contrast to the above, it is found that the
method of this embodiment can follow the actually
measured value in a wide range. 'Taking into
consideration that there is variation of about 5 to
10 mm in actually measurement accuracy of the molten
metal surface level, the molten metal surface level
detected by the method of this embodiment can be said
to be in a good correspondence with the actually
measured molten metal surface level.
[0081] As described above, this embodiment detects
the molten metal surface level while grasping the
influence of the heat transfer at the molten metal
surface position of the molten steel inside the mold
13, such as the heat removal by the powder 17. More
specifically, the position where the absolute value
of the value qy of the y-axis direction component of
the heat flux vector whose y-axis component vector is
in the direction opposite to the casting direction is
maximum, is detected as the molten metal surface
level. Accordingly, the detection accuracy of the
molten metal surface level can be increased. This
makes it possible to stably control the molten metal
surface level, and prevent overflow of the molten
steel and roll-in of suspended substance to achJ.eve
- 39 -
the improvement in the internal quality of the cast
slab. Further, this contributes to stabilization of
operation and improvement in quality such as
prevention of corrosion trouble due to local erosion
of the immersion nozzle 12 and falling of the tip of
the immersion nozzle 12, improvement in detection
accuracy of drift of the molten steel inside the mold
13 and so on.
[0082] Further, in this embodiment, a value obtained
by multiplying a value, which is obtained by
partially differentiating the
interpolation/extrapolation function· uA (x, y, t)
continuously taking values by y, by the heat
conductivity p of the material constituting the mold
13 is derived as the value q, of the y-axis direction
component of the heat flux vector. Accordingly, the
calculation accuracy of the heat flux can be
increased as compared with the case where the heat
flux is derived as a discrete value.
[0083] Further, in this embodiment, the
interpolation/extrapolation function uA (x, y, t) is
expressed by the total sum of the products of the
basis function ¢(x- Xj, y-yj, t-tj) and the weight
vector Aj. The interpolation/extrapolation function
uA(x, y, t) expressed in this way and the boundary
condition representing the balance between the heat
fluxes on the outer wall surface of the mold 13 of
the quadratic non-stationary heat conduction equation
are used as the simultaneous equations to derive the
- 40 -
weight vector l.j. Accordingly, the thermocouples in
use can be composed of only the plurality of
thermocouples arranged in one line along the y-axis
direction. This eliminates the need to arrange the
thermocouples in a plurality of lines in the x-axis
direction.
INDUSTRIAL APPLICABILITY
[0084] The present invention can' be used for
detection of the molten metal surface level of the
molten steel in the continuous casting mold.
- 41 -

CLAIMS
[Claim lJ An apparatus for detecting a molten, metal
surface level in a conti.nuous casting mold, the
apparatus comprising:
a temperature acquisition means which acquires
temper~tures measured by a plurality of temperature
measur~ment means .embedded in the continuous casting
mold along a casting direction of' the conti'nu6us
casting mold;
a heat flux derivation means which derives avalue
of a casting direction component of a heat flux
vector on an inner wall surface of the continuous
casting mold, based on a result of performing
analysis of non-stationary heat conduction inverse
problems from the temperatures derived by the
temperature acquisition means; and
a molten metal surface level derivation means
which derives a molten metal surface level inside the
continuous casting mold, based on the value of the
casting direction component of the heat flux vector
on the inner wall surface of the continuous casting
mold derived by the heat flux derivation means,
wherein the molten metal surface level derivation
means derives, as the molten metal surface level, a
position where an absolute value of the value of the
casting direction component of the heat flux vector
whose vector of the casting direction component is in
a direction opposite to the casting direction is
maximum.
- 42 -
[Claim 2] The apparatus for detecting the molten
metal surface level in the continuous casting mold
according to claim 1,
wherein the analysis of non-stationary heat
conduction inverse problems is analysis of nonstationary
heat conduction inverse problems using an
interpolation/extrapolation temperature function
satisfying a non-stationary heat "conduction equation,
and
wherein the interpolation/extrapo~ation
teiliperature function is a function uA(x, y, t)
indicating a temperature inside the continuous
casting mold at a position x in an x-axis direction
being a heat removal direction of the continuous
casting mold, a position y in a y-axis direction
being the casting direction of the continuous casting
mold, and a time t.
[Claim 3] The apparatus for detecting the molten
metal surface level in the continuous casting mold
according to claim 2,
wherein the interpolation/extrapolation
temperature function uA(x, y, t) is expressed by a
total sum of values at respective center points j of
products of a basis function ¢J decided for each
center point j and a weight vector Aj decided for each
center point j,
wherein the center point j is a point decided by
a reference position vector (Xj, YJ) indicating a
position that is a reference in the x-axis direction
- 43 -
and the y-axis direction of the continuous casting
mold and a reference time ti, which is a point 6n
three-dimensional coordinates deciided by positions in
the x-axis direction and the y-axis direction of the
continuous casting mold and a time, and
wher:·ein the basis function ¢j is a function
expressed in a form of a fundamental solution
satisfying the non-stationary he~t condricition
equation based on the center point j.
[Claim 4] The apparatus for detecting the molten
metal surface level in the contiriuous casting mold
according to claim 3,
wherein the heat flux derivation means derives a
value qy of a y-axis direction component df the heat
flux vector on the inner wall surface of the
continuous casting mold by a following (A) expression,
where following ~ is a heat conductivity of a
material constituting the continuous casting mold,
following a is a square root of a thermal
diffusion coefficient of the material constituting
the continuous casting mold,
following H (t-ti) is a Heaviside function, and
following m + 1 is a number of the center points
j .
[Expression 1]
(A)
[Claim 5] The apparatus for detecting the molten
- 44 -
metal surface level in the continuous casting mold
according to claim 3 or 4,
wherein each of a point decided by a position on
an outer wall surface of the continuous casting mold
and a time, which is a point on three-dimensional
coordinates decided by positions in the x-axis
direction and the y-axis direction of the continuous
casting mold and a time, and a po'int decided by a
position where the temperature measurement means is
embedded and a time, which is a point on threedimensional
coordinates decided by positions in the
x-axis direction and the y-axis direction of the
continuous casting mold and a time, is regarded as a
definition point of information amount,
wherein the heat flux derivation means derives
the weight vector A; by substituting the information
on the definition point of the information amount
into simultaneous equations of a boundary condition
in the non-stationary heat conduction equation and
the interpolation/extrapolation temperature function
uA(x, y, t) and solving the simultaneous equations so
as to satisfy the non-stationary heat conduction
equation, the boundary condition in the nonstationary
heat conduction equation, a thermocouple
temperature function u(x*, y*, t), and the
interpolation/extrapolation temperature function uA(x,
y, t), and derives the value qy of they-axis
direction component of the heat flux vector on the
inner wall surface of the continuous casting mold
- 45 -
using the weight vector A;,
wherein the boundary condition in the nonstationary
heat conduction equation is an expression
indicating that a heat flux based on a temperature
gradient in the x-axis direction on the outer wall
surface of the continuous casting mold ~nd on the
heat conductivity of the material constituting the
continuous casting mold is equal to a heat flux based
on a difference between a temperature on the outer
wall surface of the continuous casting mold an& a
water temperature and on a heat transfet coefficient
between the material constituting the ~ontinuous
casting mold and water,
wherein Lhe plurality of temperature measurement
means are embedded in the continuous casting mold
along the casting direction at positions different
from the outer wall surface of the continuous casting
mold, and
wherein the thermocouple temperature function
u(x*, y*, t) is a function indicating a temperature
measured by the temperature measurement means at a
position x* of the temperature measurement means in
the x-axis direction of the continuous casting mold,
a position y* of the temperature measurement means in
the y-axis direction of the continuous casting mold,
and a time t.
[Claim 6] The apparatus for detecting the molten
metal surface level in the continuous casting mold
according to claim 5,
- 46 -
wherein positions in the x-axis direction of the
plurality of temperature measurement means are same.
[Claim 7] The apparatus for detecting the molten
metal surface level in the continuous casting mold
according to claim 5 or 6,
wherein the weight vector Aj is calculated by
following (B) expression to (E) expression,
where following m is a numbei of the center
points j decided by positions on the outer wall
surface of the. continuous casting mold and times,
following l is a number of the center points j
decided by positior1s of the temperature measur~ment
means and times,
following k is an integer from 1 to m for
identifying the definition point of the information
amount,
following s is an integer from m + 1 to m + l for
identifying the definition point of the information
amount,
following j is an integer from 1 to m + 1 for
identifying the definition point of the information
amount,
following 0 is a heat conductivity of the
material constituting the continuous casting mold,
following y is a heat transfer coefficient
between the material constituting the continuous
casting mold and water,
following hs-m is a temperature measured by the
temperature measurement means,
- 47 -
following gk is a product of water temperature
and the heat transfer coefficient y between the
material constituting the continuous casting mold and
water,
following A is a (m + 1) x (m + 1) matrix,
followinr,:r [)Cl