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The Inverse Problem-Based Interfacial Heat Transfer between the Billet and Mould and the Crack Sensitiveness Area of Strand.

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Dev. Chem. Eng. Mineral Process. 14(3/4), pp. 473-486, 2006.
The Inverse Problem-Based Interfacial Heat
Transfer between the Billet and Mould, and
the Crack Sensitiveness Area of Strand
He-Bi Yin, Man Yao* and Da-Cheng Fang
Department of Materials Processing, School of Materials Science and
Engineering, Dalian University of Technoloa, Dalian I16024,
P R. China
A coupled thermo-mechanical model has been established including the finite
diference method (FDM) simulating the mould heat transfer based on the inverse
problem algorithm@om measured temperatures, and thefinite element method (FEM)
predicting the strand stress. Calculation results show that the distribution of mould
heat flux, which determines the shell thickness, corresponds to that of thermal
resistance between the mould and shell. The safety index of billet quality illustrates
that the surface cracks are likely to originate near the meniscus and propagate from
beneath the surface at the lower part of mould In this stub, the thickness
distributions of liquid and solid slag films are also calculated, showing that the heat
transfer is determined mainly by the solid slag film in the upper mould.
Numerical simulation is widely used to describe the strand thermo-mechanical
behavior and the mould heat transfer in the continuous casting process through
building mathematical models [1-4]. However, it still cannot give a precise and
comprehensive description of the various phenomena because of the complex
phenomena involved and the mutual interactions of the mould process. Often many
assumptions are made to reduce the computation time and to avoid complex problems,
e.g. intricate interfacial heat transfer from the strand to the mould.
In continuous casting process, the behavior of mould heat transfer and the
mechanical state of the strand are related. Therefore, many studies have been
conducted to develop a coupled mathematical model to calculate the thermal and
mechanical behaviors [5-71, and Thomas [ 5 ] developed a coupled 2D
thermo-mechanical model to calculate the stress distribution for slab casting. The
empirical formula describing heat flux was taken as a boundary condition in many
previous studies. However, it varies with various casting conditions [S-lo]. In order to
make the results better reflect industrial casting conditions, an inverse problem
method (based on measured temperatures from selected thermocouples buried at the
center-line) has been developed for slab and billet [ll], around the perimeter is still
* Author for correspondence (
4 73
He-Bi En,Man Yao and Da-Cheng Fang
not considered. Furthermore, currently there are no examples reported of such
research for round billet casting.
In order to make the calculation results better reflect actual casting conditions, i.e.
the non-uniformity of heat transfer around the perimeter observed in a plant [12], we
introduce an inverse problem method based on measured temperatures from the
thermocouples located at various transverse and longitudinal sections in the mould.
Attention is focused here on the heat flux, solidification state, and mechanical
behavior and even the distributions of slag film. The main results to be achieved to
reach the fixed targets are:
1. Distributions of mould heat flux and the corresponding thermal resistance.
2. Non-uniform solidified shell thickness.
3. Profiles of liquid and solid slag films thickness.
4. Safety index of cracks based on the stress in the solidified shell.
Besides, the experimental method for temperatures used in the inverse problem
algorithm is also introduced.
Temperature Data Acquisition
To obtain the real time distributions of mould temperatures and heat flux, an industrial
trial was conducted in a round billet caster machine. The molten steel from the
tundish is transferred to the mould through a straight port SEN. To measure the mould
temperatures, sheathed thermocouples with 1 mm diameter were buried in the round
billet mould (1 78 mm inner diameter, 14 mm wall thickness), and at the same time the
mould heat flux was monitored using specially designed heat flux sensor. The
thermocouples were on average 7 mm from the cold face. In six transverse sections,
whose positions were 95, 155, 245, 365, 515, 650 mm, respectively from the mould
top (expressed as L1,L2, ... L6), and six longitudinal sections, whose angles are 60"
between any two neighbors (0" stands for inner arc of the mould), there are 36
monitoring points in total. Every point has two thermocouples, having 3.5 mm
distance between them. The thermocouples lay out as adopted in the plant trial is
presented in Figure 1.
To avoid the interference of EMS on the signals, appropriate methods were taken.
The compensation lead of thermocouple and the annex cables are shielded, adopting
the insolated apparatus such as the amplifier Board, Insolated Digital Output Card and
Insolated A/D Card. In order to easily assess and decrease the effect of measurement
error, the data used in this paper are time-averaged values under the operation
parameters kept almost stable, and the time interval is 10 minutes. It was found in the
plant trial that the mould heat flux and temperatures were non-uniform around the
perimeter. Based on measured temperatures, the non-uniform heat transfer was
calculated by using an inverse problem method that can reflect the real casting
Model for Calculation
(0 The FDM/..EM coupled thetmo-mechanicalmodel
For calculating the real time thermo-mechanical behavior, the simulation model
consists of two parts. Firstly, an inverse problem algorithm is used to calculate the
4 74
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
mould heat flux with the FDM. Secondly, the coupled thermo-mechanical analysis is
based on the FEM having the same geometry and size of grids as the first part to
calculate the strand mechanical behavior based on the non-uniform heat flux obtained
from the inverse problem calculation as its boundary condition. So, a FDM/FEM
coupled thermo-mechanical analysis from experimental temperatures was developed
with its schematic diagram shown in Figure 2.
Figure 1. Thermocouples layout employed in the mould.
Figure 2. Schematic diagram of the coupled thermo-mechanical analysis.
He-Bi Yin, Man Yao and Da-Cheng Fang
(ii) Establishment of mathematical model
The following assumptions have been made according to the characteristics of heat
transfer for round billet continuous cast:
1. Mould temperature and heat flux vary around the perimeter, and heat flux is
calculated from the differential temperature by two thermocouples (3 .S mm apart)
along the radial direction. Heat flow along the drawing direction is negligible.
2. The influence of temperature on the mould physical properties is negligible below
3. The mould top and bottom are considered to be adiabatic, and heat absorbed by
the mould powder above the meniscus is also negligible.
4. The cooling water extends from the bottom to mould-wall top, and the heat
transfer coefficient at the mould/water interface is constant.
5 . The effective thermal resistance is adopted to express the complex heat transfer
between the mould and the billet.
The governing differential equation for heat conduction in the mould and molten
steel in column coordinates is:
where a = k / ( p .c P ); k is thermal conductivity; p is density; cp is specific heat;
T is temperature; r is radius; $, is radians.
To complete the mathematical description of the problem, the following initial and
boundary conditions are employed. During casting, the free liquid surface is the upper
boundary of the calculation domain.
Initial conditions:
Boundary conditions:
(2) Mouldcooling-water:
where T, and T,
are the pouring temperature and room-temperature, respectively;
r, is inner radius of mould; R~~ is effective thermal resistance between the mould
and billet; r, is surface temperature of billet; h, is convection coefficient; T, is
the temperature of cooling water; and r,,, and q,, are the temperatures of mold
outer and inner walls, respectively.
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
The R~~ between the mould and billet is unknown, so the heat transfer equation
is a morbid one. Assuming an initially uniform thermal resistance the temperature
field of the mould is calculated at the first step, and subsequently by iteratively
modifying the thermal resistance value based on the difference of the calculated and
measured temperatures. The calculation is repeated until calculated temperatures are
in agreement with measured values. This is a typical inverse heat transfer problem.
(iii) Inverse problem algorithm
Figure 3 shows the schematic diagram of longitudinal slice and the domain for inverse
heat conduction.
Figure 3. Schematic diagram showing domain for inverse heat conduction.
The temperature fields vary depending upon local thermal resistances, expressed
5 ) .when the calculated temperatures are in agreement with the
by ~ ~ ( ~Only
measurements, the given R, (?p, ) meets the requirement. If calculated temperatures
~ 5lower
are higher than the measured ones, it means the estimated ~ ~ ( are
p added;
, ) and vice versa, d ~ . ( ~ p , )
comparatively than the actual values, so ~ ~ ( ~ is
is subtracted. Given the preceding description, the profile of mould temperature is
determined by adjusting the thermal resistance between the billet and mould, ~ ~ (
by an amount, d ~ , ( ~ p according
to the following equations:
~ 5 ) ~
He-Bi fin, Man Yao and Da-Cheng Fang
where i, j = 1 , 2 , 3 ,...6 , p = 1,2,3, ...n ; ~ ~ ( q r , )is the thermal resistance of the
previous iteration; T , ~and q; are the measured and calculated temperatures
respectively; 1;; - l;,J is the differential temperature; a,,J is the iteration step.
When T; is close to T,,, and 11;; - ~ , ~&,l gthe computation stops. The E is 1°C
here. For a given R~(,J,) value, the mould temperature field is calculated. By
comparing the calculated temperatures and the measurements, RJ?;) is modified
and the temperature field is calculated again. The computation will not stop until the
calculated temperatures meet the requirement. The dichotomy method is applied to set
(iv) Calculation of solia7liquid slag jilm
A method has been developed to study the thickness of solid and liquid slag films
around the mould width based on the temperature fields of mould and strand for slab
casting. As the solidifying shell moves down through the mould, the films of solid and
liquid slag can be obtained for any transverse section of mould. This method is
described in detail in the literature [14]. Similarly, the thickness of solid and liquid
film can also be calculated, thus characterizing the non-uniformity due to the
non-uniformity of temperature around the perimeter either of mould or of strand as
discussed above.
The coupled thermo-mechanical analysis
The initial conditions applied in the FEM calculation are the same as that in the FDM
analysis already discussed, and the elastic-plastic material is considered. Furthermore,
the temperature-dependent thermal properties of the steel in both analyses were used.
An two-dimensional transient, elastic-plastic stress model was developed to determine
the internal stress in the shell arising from the change in temperature gradients
(calculated by the heat transfer analysis based on the inverse problem algorithm).
Verification of Model
The profiles of calculated and experimental heat fluxes along the drawing direction
are in agreement with each other, for either 60" or 240" longitudinal mould section as
shown in Figure 4(a). Figure 4(b) shows both profiles around the mould perimeter at
L2 under various conditions of EMS at the casting speed of 1.79 m/min, and the
calculated results are located within the various measurement ranges. The EMS has
little influence on the mould heat flux at L2, because at this location it is mainly
affected by the steel flow from the nozzle, with the exception of the obvious
4 78
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
differential heat flux at the outer-arc due to the uneven infiltration of mould flux.
Figure 5 indicates the stress profile along radii at 0.11 m below the meniscus, and the
profile accords with previously reported results [ 131. The calculated results discussed
above show that the inverse problem algorithm and the FDM @EM coupled
thermo-mechanical model are valid.
A Measurement (2404
1000 -2.
- 4500
Measurement (3.9Hz)
-A-Calculating (3Hz)
8 2000
- 2000
- 1500
- 1000
I i O ' 2iO
Position to inner-arc(degree)
(b) Along the mould circumference at L2
Figure 4. Comparison of heatJlux profles between calculatedresults and measurements.
4 79
He-BiEn, Man Yao and Da-Cheng Fang
Distance beneath surface(rn)
Figure 5. Typical temperature and stress distributions through shell thickness.
Results and Discussion
(0 Proflles of heatjlux and thermal resistance along the mould length
Heat flux and the corresponding thermal resistance are plotted against the casting
direction for various longitudinal sections in Figure 6.
2 2
- 4
2 .Y
0 6
Distance below meniscus (m)
Figure 6. Profiles of heat flux and thermal resistance along the mould length with
diflerent angles at the casting speed of 2.0 m/min.
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
The heat flux in the region 0-60 mm below the meniscus is small, and Iess than
1000 kWlm2. This is followed by an increase in heat flux until it reaches the peak
value of 3000-4500 kW/m2 in the region of 80-130 mm below the meniscus, where a
“high heat flux region” forms in which the heat flux varies over a wide range of
1500-4500 kW/m2 under various process conditions. The mould heat flux changes to
become stable along the mould height as the billet moves to the middle/lower parts of
mould, and is maintained at approximately 1000 kW/m2. It is interesting to note that
in the middle of the mould, about 280-330 mm below the meniscus, the local peak
value o f mould heat flux appears. At the mould outlet, the mould heat flux increases
gradually for some longitudinal sections. It is suggested that thermal resistance
between strand and mould may reflect the distributions of slag film thickness and gap
size between the mould and the strand, further determining the heat transfer, i.e. low
value of thermal resistance corresponding to high heat flux.
(ii) Distributions of slag firm
Figures 7(a) and (b) show the distributions of liquid and solid slag films thicknesses
calculated in the upper mould before the disappearance of the liquid slag film. In the
region from meniscus to 50 mm down the mould length, the liquid slag film is more
than 0.056 mm thick, while the solid film is almost uniform around the perimeter and
almost constant at 0.26 mm, which can explain the low and uniform heat flux near the
meniscus. In the area of 50-100 mm below the meniscus, the solid slag film is
significantly thinner, which leads to increasing heat flux along the casting direction.
The much thinner liquid slag film cannot provide sufficient lubrication, which may
result in many surface defects of the round billet. The subsequent increases of liquid
and solid slag films thickness until 200mm below the meniscus correspond to the
sharp decline of heat flux as shown in Figure 6. The maximum thickness of 2.1 mm
solid film corresponds to the lowest heat flux, while the maximum 0.2 mm thickness
of liquid film provides good lubrication. From 200 mm below meniscus to the middle
of mould, both liquid and solid slag films decrease, until at approximately 300 mm
below the meniscus all the liquid slag solidifies to become a solid slag layer.
Figure 7(c) shows that the distribution of the thermal resistance between the strand
and mould is similar to that of the solid slag film. Due to the larger thermal resistance
of solid slag film compared to the liquid slag film, the former significantly determines
the mould heat transfer in the upper mould. Hence predicting the thickness of solid
slag can help to evaluate the heat transfer in the upper mould.
(iii) Relationship between the heatflux and shell thickness
Figure 8 compares the shell thickness with heat flux at four different mould heights
below the meniscus. It shows that the mould heat flux and shell thickness vary around
the perimeter in the steady casting processes, characterizing the similar distributions
around the perimeter at any transverse section. The non-uniformity of the mould heat
flux can significantly reflect that of the shell thickness. It is can be concluded that
monitoring the heat flux can provide useful information regarding shell thickness and
He-Bi Yin, Man Yao and Da-Cheng Fang
Position to inner-atc (degree)
position to inner-arc(degree)
Figure 7. Calculated profiles at the casting speed of 2.3 m/min for: (a) liquid slag
film; (b) solid slag film; and (c) thermal resistance.
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
Position to Inner-arc (degree)
Figure 8. Comparisons of heat flux with shell thicknessfor difSerent casting times at
the casting speed of 2.3 m/min.
(iv) Safe0 index distributions of strand
A safety index of strand quality, M,, is proposed as:
M , = q / u n- 1
where oTis the temperature-dependent critical yield strength, and the values for
various temperatures are given in [ 151; and C T ~ is the calculated Von-Mises stress of
the strand.
The profiles of M , ~in the shell surface down the mould length are given in
Figure 9(a) for various casting speeds. Figure 9(a) shows that the safety index is low
in the vicinity of meniscus, followed by an abrupt increase until it reaches its
maximum value at approximately 250 mm below the mould top, and towards the
bottom of the mould it stays comparatively high. Hence, the surface cracks are prone
to occur near the meniscus. Figure 9(b) shows the variation of safety index averaged
around the perimeter along the distance from the surface of the round strand at
He-Bi Yin, Man Yao and Da-Cheng Fang
various distances below the mould top. At the initial stage of casting, about 133 mm
below the mould top, the safety index is low at the surface region. As the
solidification proceeds down to 250 mm below the mould top, the safety index at the
surface region becomes larger. However, a local minimum safety index appears just
below the surface at 647 mm mould height and the mould exit. These results indicate
that the surface defects may form at the initial stage of casting, or enlarge from
beneath the surface at the lower part. It can also be deduced that the safety indices are
comparatively low in the mushy zone at any distance from the mould top, and the
internal cracks tend to form at the solidieing front.
65 x 4-
P.- .
a0 3 -3210
600 700
Distance from mould top (m)
2 3 4 5 6 7 8 9 1 0 1 1 1 2
Distance beneath surface(m)
Figure 9. The calculated distributions of safety index o j (a) the shell surface
averaged along mould circumferencefor direrent casting speeds; (b) at various
distances from mould top at the casting speed of 2.3 d m i n .
Heat Transfer between Billet and Mould, and Crack Sensitiveness Area of Strand
Based on an inverse problem method, a FDM/FEM coupled thermo-mechanical
model was developed in order to calculate the thermal and mechanical behaviors that
reflect the industrial continuous casting process of round billet. The conclusions are as
1. The mould heat flux and the shell thickness vary around the perimeter, and the
profiles of mould heat flux can reflect that of the shell thickness to some extent.
2. Non-uniform distributions of liquid and solid slag films thicknesses are calculated,
and the solid slag film determines the mould heat transfer to a significant extent in
the upper mould.
3. By means of the safety index predicting the strand quality, the surface cracks are
prone to originate near the meniscus or propagate from beneath the billet surface
at the lower part of mould, and the internal cracks may initiate in the mushy zone.
The authors gratefully acknowledge the financial support of NSFC-Steel & Iron
Union Research Fund and Science & Technology Research Project of MOE (Ministry
of Education) of China. The authors would like to thank Dr. Liu Xiao and Dr. Yu Yan
for their dedication and efforts in conducting the plant trials in Baosteel Co.
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base, mould, strany, interfacial, problems, heat, area, transfer, sensitiveness, inverse, crack, billets
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