Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method
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Інститут прикладної математики і механіки НАН України
2008
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| Cite this: | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method / G.O. Ivanova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 61-73. — Бібліогр.: 4 назв. — англ. |
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| citation_txt | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method / G.O. Ivanova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 61-73. — Бібліогр.: 4 назв. — англ. |
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ISSN 1683-4720 Труды ИПММ НАН Украины. 2008. Том 17
UDK 681.5:51-74
c©2008. G.O. Ivanova
IDENTIFICATION OF CONVECTIONHEAT TRANSFER COEFFICIENT
OF SECONDARY COOLING ZONE OF CCM BASED ON LEAST
SQUARES METHOD AND STOCHASTIC APPROXIMATION METHOD
The detailed mathematical model of heat and mass transfer of steel ingot of curvilinear continuous
casting machine is proposed. The process of heat and mass transfer is described by nonlinear partial
differential equations of parabolic type. Position of phase boundary is determined by Stefan conditions.
The temperature of cooling water in mould channel is described by a special balance equation. Boundary
conditions of secondary cooling zone include radiant and convective components of heat exchange and
account for the complex mechanism of heat-conducting due to airmist cooling using compressed air
and water. Convective heat-transfer coefficient of secondary cooling zone is unknown and considered as
distributed parameter. To solve this problem the algorithm of initial adjustment of parameter and the
algorithm of operative adjustment are developed.
Introduction. Improved computing significantly increased role of mathematical
modeling in research of thermo-physical processes. This, in turn, imposes stricter re-
quirements to accuracy and efficiency of mathematical models.
It is well known that successful modeling mostly depends on the right choice of
a model, which is directly affected by reliability of thermo-physical parameters used.
Frequently, empirical data alone cannot be an exhaustive source of the information on
conditions of solution uniqueness.
Therefore, recently particular attention is being paid to the solution of inverse prob-
lems of heat conduction which allow determining to define thermo-physical properties
(like boundary conditions) of an object based on some known (sometimes limited) infor-
mation about the temperature field. The main challenge in development of mathematical
models of technological processes is selection of process parameters.
Determining some thermal or physical parameters is necessary each time an industrial
process is being modeled. In particular, convective heat-transfer coefficient (CHTC) on
a surface of an ingot in the secondary cooling zone which depends on multiple factors.
Value of CHTC can also vary based on time and on space coordinates. Thus, there is a
problem of identification of the CHTC as distributed parameter.
In the given work algorithms of initial adjustment of parameter when at the disposal
of there is enough plenty of points in which the temperature on a surface of an ingot
is measured, and operative adjustment when the temperature is measured only in one
point on a surface are considered.
1. Statement of problem. The thermal field of the moving steel ingot and mold
wall in the system of coordinates attached to motionless construction of CCM is consid-
ered [1]. In fig.1 the diagram of CCM is introduced.
The heat conduction in the steel ingot in the mold area is described by nonstationary,
61
G.O. Ivanova
Figure 1.
nonlinear heat and mass transfer equation:
∂T (τ, x, z)
∂τ
+ v(τ)
∂T (τ, x, z)
∂z
=
=
1
c(T, x, z)ρ(T, x, z)
{
∂
∂x
[
λ(T, x, z)
∂T
∂x
]
+
∂
∂z
[
λ(T, x, z)
∂T
∂z
]}
,
0 < x < l, 0 < z < Z
(1)
and the boundary conditions:
−λ(T, x)
∂T
∂z
= 0, 0 ≤ x ≤ l,
∂T
∂x
∣∣∣∣
x=0
= 0, 0 ≤ z ≤ Z,
λ(T, z)
∂T
∂x
∣∣∣∣
x=l
=
λgz
δ
(
T |x=l+δ − T |x=l
)
+ σn
[(
T |x=l+δ
100
)4
−
(
T |x=l
100
)4
]
,
0 ≤ z ≤ Z,
(2)
where v(τ) – withdrawal rate, 2l – ingot thickness, Z – height of ingot in the mould,
62
Identification of Convection Heat Transfer Coefficient
T (τ, x, z) – metal temperature, c(T, x, z) – metal specific heat, ρ(T, x, z) – density,
λ(T, x, z) – thermal conduction, δ – effective thickness of air gap between ingot and
the mould wall, λgz – thermal conduction coefficient of gap gas mixture, T |x=l – surface
temperature of the ingot, T |x=l+δ – surface temperature of mold wall, σn – the resulted
radiation coefficient.
Conditions of equality of temperatures and Stefan conditions, and also boundary and
initial conditions for the phase boundary are set:
T (τ, x, z)|x=ξ−(τ,z) = T (τ, x, z)|x=ξ+(τ,z) = Tkr,
λ(T, x, z)
∂T
∂n̄
∣∣∣∣
x=ξ−(τ,z)
− λ(T, x, z)
∂T
∂n̄
∣∣∣∣
x=ξ+(τ,z)
= µρ(Tkr)
(
∂ξ
∂τ
+ v · ∂ξ
∂z
)
,
0 ≤ z ≤ Z,
ξ(τ, 0) = l, ξ(0, z) = ξ0(z),
(3)
where ξ – the phase boundary function of two variables x = ξ(τ, z), µ – crystallization
latent heat, Tkr – crystallization temperature (average of the interval “liquidus – solidus”),
n̄ – normal to the boundary of phases.
Heat equation for mould walls:
∂T (τ, x, z)
∂τ
=
1
c(T, x, z)ρ(T, x, z)
{
∂
∂x
[
λ(T, x, z)
∂T
∂x
]
+
∂
∂z
[
λ(T, x, z)
∂T
∂z
]}
,
z0 < z < Z, l < x < d
(4)
Boundary conditions for mould walls represent the character of heat exchange on
each sight of wall:
λ(T, z)
∂T
∂x
∣∣∣∣
x=d
= α1 (Twater(τ, z)− T |x=d) , z0 ≤ z ≤ Z,
λ(T, x)
∂T
∂z
∣∣∣∣
z=Z
= α2 (Tos.2 − T |z=Z) , l ≤ x ≤ d, z = Z,
− λ(T, x)
∂T
∂z
∣∣∣∣
z=z0
= α3
(
Tos.3 − T |z=z0
)
, l ≤ x ≤ d, z = z0,
λ(T, z)
∂T
∂x
∣∣∣∣
x=l+δ
=
=
λgz
δ
(
T |x=l+δ − T |x=l
)
+ σn
[(
T |x=l+δ
100
)4
−
(
T |x=l
100
)4
]
,
0 ≤ z ≤ Z, x = l + δ,
−λ(T, z)
∂T
∂x
∣∣∣∣
x=l+δ
= α4 (Tos.1 − T |x=d) + Cn
[(
Tos.1
100
)4
−
(
T |x=d
100
)4
]
,
z0 ≤ z ≤ 0, x = l + δ,
(5)
63
G.O. Ivanova
where d – mold wall thickness, z0 – mold wall altitude over meniscus level, α1 – heat
transfer coefficient from the mould wall to cooling water, Twater(τ, z) – cooling water
temperature in the mold channel, α2,3,4 – heat transfer coefficients from other mould
wall to environment, Tos.2,3,4 – environment temperature, Cn – the resulted radiation
coefficient.
The following balance equation describes distribution of cooling water temperature
in the mold channel:
c ·S · vwater
∂Twater(τ, z)
∂z
= PIα1 (Twater(τ, z)− T |x=d)−PEαE (Twater(τ, z)− TE) , (6)
where c – volume heat capacity of water, S – the cross-section area of the mold channel,
vwater – water velocity, PI – perimeter of the interior mold wall, PE – perimeter of the
external mold wall, αE – heat transfer coefficient from cooling water to the external
mould wall, TE – external mould wall temperature.
The cooling water temperature on the entry in the mould channel is known:
Twater(0, Z) = Twater1(τ) (7)
and it’s initial distribution in the mold channel:
Twater(0, z) = Twater0(z) (8)
The following equation describes heat and mass transfer on the curvilinear sections
of CCM:
∂T
∂τ
+ θm(τ)
∂T (τ, r, ϕ)
∂ϕ
=
1
c(T, r, ϕ)ρ(T, r, ϕ)
×
×
{
∂
∂r
(
λ(T, r, ϕ)
∂T
∂r
)
+
1
r2
· ∂
∂ϕ
(
λ(T, r, ϕ)
∂T
∂ϕ
)
+
λ(T, r, ϕ)
r
· ∂T
∂r
} (9)
where θm – angular velocity of ingot driving on the m-th curvilinear section.
The conditions for unknown boundary on the curvilinear sections are
T (τ, r, ϕ)|r=ξ1,2−(τ,ϕ) = T (τ, r, ϕ)|r=ξ1,2+(τ,ϕ) = Tkr,
λ(T, r, ϕ)
∂T
∂n̄
∣∣∣∣
ξ1−
− λ(T, r, ϕ)
∂T
∂n̄
∣∣∣∣
ξ1+
= µρkr
(
θm(τ) · ∂ξ1
∂ϕ
+
∂ξ1
∂τ
)
,
ξ1(0, ϕ) = ξ10(ϕ),
λ(T, r, ϕ)
∂T
∂n̄
∣∣∣∣
ξ2+
− λ(T, r, ϕ)
∂T
∂n̄
∣∣∣∣
ξ2−
= −µρkr
(
θm(τ) · ∂ξ2
∂ϕ
+
∂ξ2
∂τ
)
,
ξ2(0, ϕ) = ξ20(ϕ),
(10)
64
Identification of Convection Heat Transfer Coefficient
where ξ1(ϕ) and ξ2(ϕ) – phase boundaries (interfaces).
The boundary conditions of the secondary cooling zone include radiant and convective
components of heat exchange and account for the complex mechanism of heat-conducting
due to air-mist cooling using compressed air and water. The boundary conditions on the
curvilinear sections are
− λ(T, ϕ)
∂T
∂r
∣∣∣∣
r=rm
= αI(Gm(τ), ϕ) · (TIm −T |r=rm
)
+ CIm
(
T 4
Im
− (T |r=rm
)4
)
(11)
λ(T, ϕ)
∂T2
∂r
∣∣∣∣
r=rm+2l
=
= αE(Gm(τ), ϕ) · (TEm − T |r=rm+2l
)
+ CEm
(
T 4
Em
− (T |r=rm+2l)
4
)
,
(12)
where αI(Gm(τ), ϕ), αE(Gm(τ), ϕ) – convective heat transfer coefficients, CIm , CEm –
the resulted radiation coefficients, TIm , TEm – environment temperatures, Gm(τ) – water
discharge on the m-th section.
The following equation describes the heat and mass transfer on rectilinear sections of
CCM (analogously (1)):
∂T
∂τ
+ v(τ)
∂T (τ, x, z)
∂x
=
=
1
c(T, x, z)ρ(T, x, z)
{
∂
∂x
[
λ(T, x, z)
∂T
∂x
]
+
∂
∂z
[
λ(T, x, z)
∂T
∂z
]} (13)
When the liquid phase passes the straightening point on the rectilinear section of the
secondary cooling zone, the conditions for the unknown phase boundary are set:
T (τ, x, z)|x=ξ1,2−(x,z) = T (τ, x, z)|x=ξ1,2+(x,z) = Tkr,
λ(T, x, z)
∂T
∂n̄
∣∣∣∣
ξ1−
− λ(T, x, z)
∂T
∂n̄
∣∣∣∣
ξ1+
= µρkr
(
v(τ) · ∂ξ1
∂x
+
∂ξ1
∂τ
)
,
λ(T, x, z)
∂T
∂n̄
∣∣∣∣
ξ2+
− λ(T, x, z)
∂T
∂n̄
∣∣∣∣
ξ2−
= −µρkr
(
v(τ) · ∂ξ2
∂x
+
∂ξ2
∂τ
)
.
(14)
The boundary conditions for the rectilinear section:
−λ(T, x)
∂T
∂z
∣∣∣∣
z=zp
= αI(Gm(τ), x) ·
(
TI − T |z=zp
)
+ CI4
(
T 4
I − (T |z=zp
)4
)
λ(T, x)
∂T
∂z
∣∣∣∣
z=zp+2l
=
= αE(Gm(τ), x) · (TE − T |z=zp+2l) + CE4
(
T 4
E − (T |z=zp+2l)
4
)
.
(15)
65
G.O. Ivanova
We assume, that the thermal stream of the end of the rectilinear site is equal to zero:
λ(T, z)
∂T
∂x
∣∣∣∣
x=xf
= 0. (16)
The initial conditions for entire temperature field (on the rectilinear and curvilinear
sections):
T (0, x, z) = T0(x, z)
T (0, r, ϕ) = T0(r, ϕ).
(17)
It is required to define the convective heat transfer coefficients αI(Gm(τ), ϕ), and
αE(Gm(τ), ϕ) using the available information about ingot temperature.
This is a boundary inverse problem and it is ill-posed in classical sense. Well-
posedness in classical sense (or Hadamard well-posedness) means performance of three
conditions: an existence of a solution, its uniqueness and stability (input data continu-
ous dependence). In our case the third condition is not satisfied. That is easy to verify
using for the solution this problem the method of direct reversion [2]. Therefore other
approaches are necessary to solve this problem.
2. CHTC identification by least squares method. Consider an ingot in first
cooling section of secondary cooling zone. We have ingot surface temperature mea-
surements in some points. So we have to solve the Dirichlet problem for interior heat
exchange. The finite-difference method was used to approximate the solution of this
problem. The convective heat-transfer coefficient (CHTC) has special distribution along
the surface of the ingot. Parabolic function with a sufficient degree of accuracy approxi-
mates distribution of CHTC on the part of surface that is exposed to water-air spraying
from one nozzle. This parabola has maximal value in the point that corresponds to nozzle
coordinate. CHTC is considered as constant on the parts of the surface not subjected to
the forced cooling (fig.2).
Figure 2.
In one cooling section the same type spray nozzles are installed. They give an identical
water-air spray. Hence the CHTC is the same parabola shifted along the abscissa axis
(fig.2).
66
Identification of Convection Heat Transfer Coefficient
All sites under spray nozzles can be reduced to the coordinate origin so that the peak
of each parabola should be over the coordinate origin. Hence, it is necessary to define
only two parameters – αp and αc. So, α(ϕ) is given by
α(ϕ) = αc − αp
w2
ϕ2 + αp. (18)
Consider the parts of the section, on which α(ϕ) = αc = const. Let K be the
ensemble of points ϕi, in which CHTC is equal to constant. Let B be the ensemble of
other points.
The finite-difference approximation of boundary condition (11) is
λi,0
Ti,2 − 4Ti,1 + 3Ti,0
2q
= αc(TI1 − Ti,0) + CI1(T
4
I1 − T 4
i,0), (19)
where q – step of finite-difference grid by radius r1 [3].
It follows that the discrepancy of heat flows on the boundary is:
∆ = λi,0
Ti,2 − 4Ti,1 + 3Ti,0
2q
− CI1
(
T 4
I1 − T 4
i,0
)− αc (TI1 − Ti,0) .
Let us denote
Pi = λi,0
Ti,2 − 4Ti,1 + 3Ti,0
2q
− CI1
(
T 4
I1 − T 4
i,0
)
, Qi = TI1 − Ti,0.
Then we find a value αc, such that the sum of squares of discrepancies is minimum,
i.e. the follow condition is satisfied
S =
∑
i
(Pi − αcQi)2 → min, ∀i : ϕi ∈ K.
A necessary condition of the extremum existence of the function S(αc) is:
∂S
∂αc
= −2
∑
i
Qi(Pi − αcQi) = 0.
It follows that
αc =
∑
i
QiPi
∑
i
Q2
i
.
To the each point ϕi from В we will put in conformity a point yi on the seg-
ment [−w, w] such that |yi| is equal to the distance from the corresponding ϕi to the
coordinate of the nearest spray nozzle. From (18) and (19) we gain a discrepancy
∆ = λi,0
Ti,2 − 4Ti,1 + 3Ti,0
2q
− CI1
(
T 4
I1 − T 4
i,0
)−
(
αc − αp
w2
y2
i + αp
)
(TI1 − Ti,0) .
67
G.O. Ivanova
Then we can find a value αp, such that the sum
S =
∑
i
(Pi − (αc − αp
w2
y2
i + αp) ·Qi)2 → min .
From the following necessary condition of extremum existence
∂S
∂αp
= 2
∑
i
(
Pi −
(
αc − αp
(
y2
i
w2
− 1
))
Pi
)(
Qi
(
y2
i
w2
− 1
))
= 0
we obtain αp
αp =
αc
∑
i
Q2
i
(
y2
i
w2
− 1
)
−
∑
i
PiQi
(
y2
i
w2
− 1
)
∑
i
Q2
i
(
y2
i
w2
− 1
)2 .
On fig.3 comparative results of calculations (1 – by the method of direct reversion,
2 – by the least squares method) are presented. For steel grade st40, width of a slab
is 1m, l = 0,1m and v = 1(m/minute). The decision obtained by the method of direct
reversion is unstable and unsuitable for practical use. The second curve represents a spline
approximation, which is gained as a result of the decision of a problem of identification
by the least squares method.
Figure 3.
68
Identification of Convection Heat Transfer Coefficient
Thus, we fined the spline approximation of the CHTC, which is distributed on the
surface of the moving ingot. This approximation gives the minimum of mean-square
deviation between measured surface temperature and calculated one according to the
model as the result of solving of the direct problem. The CHTC for other sections of the
secondary cooling zone is analogously defined. It should be noted that an advantage of
the offered method is that the estimation error of the least squares method is negligibly
small by relatively small number of abnormal measurements. It is very important in case
of temperature measurement of a partially oxide scaled ingot surface.
3. Operative adjustment of convective heat transfer coefficient (CHTC).
CHTC obtained by initial adjustment varies under changes of various parameters of
process (for example, ambient temperatures). Therefore, it is necessary to provide its
operative adaptation during work CCM. The fine-tuning of parameters should be carried
out in real time. But during usual work of CCM the information on a thermal condition
of an ingot is limited to temperature indications in small number of points of the surface
of an ingot. Such algorithms can be based on the stochastic approximation method [4].
The temperature on the ingot surface is measured in every equal small time intervals.
Let us denote the measuring temperature data T ∗j . The computer models the casting pro-
cess using the presented mathematical model. The under model calculated temperature
in the corresponding point we denote by Tj . It is necessary to correct the model param-
eters using information about deviations between measured and calculated temperature
data to reduce these deviations to minimum. The difficulty of the decision of the given
problem is that temperature measurements are deformed by a random telemetry error.
Operative fine-tuning consists in refinement of the constant value αc, which defines
the distribution of the convective heat transfer coefficient obtained by the solving of the
problem of the initial adjustment of parameters.
For using the algorithm of stochastic approximation it is necessary, that the random
error of temperature indications would have the zero average and the finite variance.
The algorithm of parameter adjustment is
αj+1 = αj − kj(T ∗j − Tj), (20)
where αj – j-th approximate value of αc, kj – special sequence of numbers, which satisfies
to the following conditions:
lim
j→∞
kj = 0,
∞∑
j=1
kj = ∞,
∞∑
j=1
(kj)2 < ∞. (21)
For example the following elementary sequence satisfies to such conditions
kj =
a
b + j
,
where a, b ∈ R, a > 0. Selecting numbers a and b, and also other sequences satisfying to
the conditions (21), it is possible to change speed of convergence of algorithm. In [3], for
example, it is recommended to keep kj as constant while the sign of discrepancy T ∗j −Tj
not vary, and change then kj so that to satisfy to above mentioned restrictions.
69
G.O. Ivanova
Truncation condition of the parameter fine tuning algorithm work is occurrence of m
last received approximations αn+1, αn+2, . . . , αn+m in a vicinity of αn serves:
|αn − αn+i| < ε, ∀i = 1, . . . , m.
If the condition is executed, assume αc is equal αn. For check we use values CHTC
which have been picked up experimentally at the decision of a direct problem of modeling
of thermal field CCM [1].
4. Examples of realization of the stochastic approximation method. Nu-
merical modeling allows establishing the basic features of trajectories of parameter fine-
tuning process. On fig.4 trajectories of parameter fine-tuning, characterizing a deviation
of the distributed parameter from true value, for the algorithm using sequence
kj =
a
j
, j = 1, 2, 3, ...
are presented at various values of factor a. When a < 1 very slow convergence is observed.
In this case the time of parameter tuning is inadmissible big.
Figure 4.
We assume that a = 1 then value of the parameter is in enough small vicinity of true
value approximately after 200th iteration. At a = 2 the trajectory of parameter fine-
tuning reflects oscillations with damped amplitude and frequency and not later than for
70
Identification of Convection Heat Transfer Coefficient
200 iterations the parameter is adjusted. At increase a > 2 the amplitude of oscillations
grows. In this case also oscillations with damped amplitude and frequency are observed,
but for fine-tuning it is required considerably more iterations.
From here we conclude, that for the chosen sequence the best values of the factor a
is a number from interval 1 ≤ a ≤ 2.
Now we investigate now influence of value b on speed of the algorithm’s convergence.
On fig.5 trajectories of parameter fine-tuning are shown for various values of b. Values b
less than zero lead to that fine-tuning go in a "wrong" direction while the denominator
is negative and at i = −b the denominator is equal to zero. Increase of b leads to
decrease of a velocity of convergence of algorithm. The same results have been obtained
for sequences, which will be described below. Therefore further parameter b everywhere
will be chosen to be equal to zero.
Figure 5.
The following sequence also satisfies to conditions (21)
kj =
a
nj
, nj+1 =
{
nj , (T ∗j − Tj)(T ∗j+1 − Tj+1) > 0
j + 1, (T ∗j − Tj)(T ∗j+1 − Tj+1) ≤ 0
. (22)
Results of this algorithm execution are presented on fig.6. In this case factor a needs
to be chosen between 1 ≤ a ≤ 3. Values out of this range give smaller speed of algorithm
convergence.
71
G.O. Ivanova
Consider another sequence, which also satisfies to conditions (21)
kj =
a
nj
, nj+1 =
{
nj , (T ∗j − Tj)(T ∗j+1 − Tj+1) > 0
nj + 1, (T ∗j − Tj)(T ∗j+1 − Tj+1) ≤ 0
. (23)
It has slower convergence than the previous two sequences. Results of calculations
with use of this sequence are presented on fig.7. Factor a can be chosen between 0.5 ≤
a ≤ 2. And, if 1.2 ≤ a ≤ 1.5, than obtained approximations differ from the true value
no more than on 6 % after 20 iterations already.
Figure 6.
In the conclusion it is necessary to add, that the advantage of stochastic approxima-
tion algorithm is its successful application for wide enough range of initial values of the
tuned parameter.
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Continuous Casting Machine. – Electronic Modeling – 2008.–– Vol.30. – №3. – P.87-103. (in russian).
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3. Marchuk G.I. Methods of computational mathematics. – Moskow: Nauka, 1980, 535p. (in russian).
72
Identification of Convection Heat Transfer Coefficient
Figure 7.
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1971, New York and London.
Institute of Applied Mathematics and Mechanics of NASU, Donetsk
ivanova@iamm.ac.donetsk.ua
Received 07.11.08
73
содержание
Том 17
Донецк, 2008
Основан в 1997г.
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| id | nasplib_isofts_kiev_ua-123456789-20010 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1683-4720 |
| language | English |
| last_indexed | 2025-11-30T13:49:37Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Ivanova, G.O. 2011-05-20T07:33:08Z 2011-05-20T07:33:08Z 2008 Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method / G.O. Ivanova // Труды Института прикладной математики и механики НАН Украины. — Донецьк: ІПММ НАН України, 2008. — Т. 17. — С. 61-73. — Бібліогр.: 4 назв. — англ. 1683-4720 https://nasplib.isofts.kiev.ua/handle/123456789/20010 681.5:51-74 en Інститут прикладної математики і механіки НАН України Труды Института прикладной математики и механики НАН Украины Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method Article published earlier |
| spellingShingle | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method Ivanova, G.O. |
| title | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method |
| title_full | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method |
| title_fullStr | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method |
| title_full_unstemmed | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method |
| title_short | Identification of Convection Heat Transfer Coefficient of Secondary Cooling Zone of CCM based on Least Squares Method and Stochastic Approximation Method |
| title_sort | identification of convection heat transfer coefficient of secondary cooling zone of ccm based on least squares method and stochastic approximation method |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/20010 |
| work_keys_str_mv | AT ivanovago identificationofconvectionheattransfercoefficientofsecondarycoolingzoneofccmbasedonleastsquaresmethodandstochasticapproximationmethod |