http://web.mst.edu/~lekakhs/webpage%20Lekakh/ Methodologies of Inverse Simulation of Metallurgical Processes. A Review Simon N. Lekakh Missouri University of Science and Technology 223 McNutt Hall, 1400 N. Bishop Street, Rolla, MO 65409, USA Synopsis Inverse simulation is a powerful research technique with a long and successful history of modeling metallurgical processes. However, the capabilities of this method are underestimated because only a few ‘ready to use’ simulation packages have an embedded in verse simulation option. Inverse simulation in metallurgical research is not a standalone tool, but works in combination with other experimental and direct simulation methods. There is no one unified rule for using inverse simulation for metallurgical research. This article describes several approaches developed for obtaining transient properties of materials used in metal casting processes, simulating steelmaking processes in the ladle, tundish and continuous caster, and reconstructing solidification kinetics based on observed final cast structure. In each described case, the Characteristic Functions were determined; these Characteristic Functions play a key role in inverse optimization. In described cases, the combinations of direct CFD method and specially designed experiments were used together with inverse simulations. This review shows a variety of practical applications of inverse simulation in metallurgy, rather than providing detailed mathematical analysis of these methods. Key words: inverse simulation, metallurgical processes, ladle, tundish, caster, ceramic shell, solidification Introduction Inverse simulations are a powerful research technique for studying metallurgical processes as an intermediate between direct numerical simulations and experimental methods. In direct simulations, the governing equations, boundary conditions, and the properties are given at the start of modeling. Using the direct simulation techniques a model output response is determined for a given set of initial conditions and for selected input variables. The quality of the direct simulation is judged by its correspondence with metallurgical experiment. However, in high temperature metallurgical processes, the disagreement of simulated predictions with experimental measurement could be related to: (i) unknown phenomena which were not considered in the basic model, (ii) poorly understood boundary conditions where high-temperature gradient and multiple physicochemical interactions take place, and (iii) transient material properties affected by a high-temperature process history. Therefore in some cases, the agreement between direct simulation of metallurgical processes and experiment could be unsatisfactory. In these conditions, an inverse simulation allows the investigator to obtain the best agreement between modeled and experimental results. Inverse simulations are defined as the reverse of direct simulation methodology, where the time history of output variables is prescribed before simulations and the inverse simulation algorithm allows the investigator to determine the time history of the corresponding input variables. In metallurgical processes, the inverse simulation approach is of practical value for a number of reasons. This review provides several practical examples of inverse simulation techniques used by the researches to obtain transient high temperature properties of materials used in metal casting processes, to verify boundary conditions, to simulate steelmaking processes in the ladle, tundish, and continuous caster; and to reconstruct solidification kinetics based on the observed final cast structure. In these cases, the combination of direct CFD methods and specially designed experiments were used together with inverse simulation. The goal of this review is to show rather a variety of practical applications of inverse simulation
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http://web.mst.edu/~lekakhs/webpage%20Lekakh/
Methodologies of Inverse Simulation of Metallurgical Processes. A Review
Simon N. Lekakh
Missouri University of Science and Technology
223 McNutt Hall, 1400 N. Bishop Street, Rolla, MO 65409, USA
Synopsis
Inverse simulation is a powerful research technique with a long and successful history of modeling
metallurgical processes. However, the capabilities of this method are underestimated because only a few
‘ready to use’ simulation packages have an embedded in verse simulation option. Inverse simulation in
metallurgical research is not a standalone tool, but works in combination with other experimental and direct
simulation methods. There is no one unified rule for using inverse simulation for metallurgical research.
This article describes several approaches developed for obtaining transient properties of materials used in
metal casting processes, simulating steelmaking processes in the ladle, tundish and continuous caster, and
reconstructing solidification kinetics based on observed final cast structure. In each described case, the
Characteristic Functions were determined; these Characteristic Functions play a key role in inverse
optimization. In described cases, the combinations of direct CFD method and specially designed
experiments were used together with inverse simulations. This review shows a variety of practical
applications of inverse simulation in metallurgy, rather than providing detailed mathematical analysis of
Solidification characteristics of the real industrial alloy can also be determined from the cooling curves
obtained from the industrial castings. A single thermocouple method with Newtonian analysis4,5) or a two
thermocouples method with Fourier analysis6) were suggested. However, simplified analytical assumptions
of thermal field used in these methods decrease accuracy of the calculated solidification characteristics.
Methodology. Commercial software Procast7) and Magmasoft8) have the special modules for inverse
simulation of industrial metal casting trial. The energy conservation equation is used for determination of
the value of latent heat (L) and the function of solid fraction (fs) vs temperature (T) or solidification time (τ)
for fixed values of solid and liquid heat capacities (c), casting-mold boundary conditions, and mold thermal
properties:
𝑝(𝑐 − 𝐿𝑑𝑓
𝑑𝑇)𝜕𝑇
𝜕𝜏= ∇(𝑘∇𝑇) (1)
The experimental cooling curve obtained from the casting T = Ψ(τ) is used to develop the Characteristic
Function, which could be cooling rate vs solidification time (dT/dτ = Θ(τ)) or cooling rate vs solidification
temperature (dT/dτ = Φ(T))9). The Φ(T)) function better indicates the solidification end. Inverse CFD
simulation was used to fit the experimental and the virtually simulated Characteristic Functions by varying
the latent heat (L) value and the incremental value of solid fraction ( df/dT).
Solidification path and thermo-physical properties. The solidification passes of three high alloyed
stainless steels and two nickel based alloys were determined from the several experimental cooling curves
obtained from each casting9). Initial material properties were generated using thermodynamic software and
this data set was modified using an interactive inverse method. The used method utilizes comparison
between the measured cooling rate with the corresponding simulated value to direct changes in the dataset,
until satisfactory agreement between simulated and measured Characteristic Functions was reached.
Authors9) mentioned that the presented iterative inverse method was more accurate than traditional
Newtonian and Fourier thermal analysis.
2. Optimization of interfacial metal-mold heat transfer coefficient (IHTC)
Problem statement. The casting simulation predictions are only as accurate as the material and processing
properties used as input. The casting experiments demonstrated the large variations of casting-mold
boundary conditions in different casting processes which cannot be accurately obtained from pure
theoretical models. A number of studies have been conducted to determine interfacial heat-transfer
coefficient (IHTC) between a solidified casting and a mold, which can be sand or permanent and poured
by gravity or assisted by low pressure or high pressure10,11) . IHTC play important role in productivity of
steel continuous casting process12). Analysis of effective IHTC (heff) having convection and radiation
components is based on the following equation:
q=heff (Tcasting-Tmold) (2)
where: q is in the heat flux, Tcasting and Tmold are the temperatures at the casting and mold surfaces.
Since many factors play role in heff, determining it accurate value is a very specific to a given casting shape
and the cast process.
Methodology. Inverse simulation is a common method to determine heff for the specific casting conditions.
For example, to study IHTC in low pressure aluminum casting process, a set of thermocouples was inserted
into the casting cavity and the wall of permeant mold to collect the thermal curves10). A OptCast module of
ProCAST finite-element simulation software7) was used to generate the virtual cooling curves (Tsim). These
Characteristic Function was used in inverse simulation of heff to minimize an objective error function (𝜑) by fitting the simulated (Tsim) and experimental (Texp) cooling curves:
𝜑 = ∑ ∑ (𝑇𝑠𝑖𝑚𝑀𝑗=1
𝑁𝑖=1 − 𝑇𝑒𝑥𝑝)2 (3)
where: N is the number of thermocouples and M is the number of time-steps.
Interfacial heat-transfer coefficient. The described method with some variations was used to determine
IHTC for different metal casting processes10,11). In low pressure process, a large difference between INTC
for concave and convex surface of aluminum casting solidified in permeant mold was observed10).
Knowledge about IHTC is critically important for continuous steel casting processes12,13). The overall heat
flux from the strand to the mold can be measured from the heat flux passed to cooling water. The mold
instrumented by thermocouples is used for the direct measurement of the local heat flux13,14). These
measured heat fluxes is used as Characteristic Function for inverse determination of IHTC values in
continuous casting processes.
3. Transient thermal properties of investment ceramic shells
Problem statement. Broadly used in metal casting, investment casting shells consist of amorphous silica
binder and ceramic aggregates. The shell is subject to the firing and pouring heating/cooling cycles during
the investment casting process. Amorphous silica binder devitrification and polymorphic transformations
in the aggregate affect the investment shell thermal properties and depend on thermal history. Reliable and
realistic thermal properties data for investment casting shell molds are required to correctly simulate casting
solidification and predict the shrinkage.
Methodology. Inverse simulations of casting trials have been used often to obtain the thermal properties of
cast alloys and mold materials, as well as the specific thermal conditions at the casting/mold boundary15,16).
The experimental cooling curves obtained from the casting or the mold (or both) are used as the
Characteristic Function. Inverse CFD simulation by varying the thermal properties input is used to fit the
experimental and the virtually simulated cooling curves. This approach is relative simple; however, when
unknown cast material properties are inverse simulated from the experiment by postulating the mold
properties and the boundary conditions or vice versa, large uncertainties result. To avoid these possible
errors, the experiment must be designed carefully to minimize the number of unknown variables. An
example is the pouring of pure molten nickel16,17), with well documented thermal properties, into ceramic
shell mold to determine the transient thermal properties of the mold material (Fig. 1). The experiments were
improved by (i) covering the mold with low-conductivity ceramic fiber to decrease external heat flux, and
(ii), the thermal properties determined from a steady state laser flash method were used for stating input in
inverse simulations. CFD inverse simulation was done using the optimization module of MAGMASOFT®8).
Figure 1. Experimental and fitted inverse simulated cooling curves for casting (pure Ni) and a ceramic
shell mold [16].
Thermal properties. Fig. 2 compares thermal property data for ceramic shell material obtained by the
inverse method and by laser flash diffusivity experiments16). It was observed that the thermal conductivity
values were fairly similar for the two methods; however, the inverse method determined higher values of
heat capacity than the laser flash method, due to the effect of reaction heat released during shell sintering
and devitrification. Combining the experimental results and those calculated using inverse simulation a
comprehensive thermal properties database for seven industrial shell systems was developed 16,17).
Figure 2. Comparison of heat capacity (Cp) and coefficient of thermal conductivity (K) of silica-based
ceramic investment shell determined by inverse simulation and laser flash methods [16].
4. Simulations of Steelmaking Processes
Problem statement. From a practical standpoint, the major objectives in the study of liquid steel
processing can be classified into three levels: (1) process understanding, (2) process design, and (3)
process control. Multi-phase CFD simulations have been intensively used to study the melt flow
phenomena and chemical reactions19); however, such CFD simulations are time-consuming and too slow
for application to on-line process control (Fig. 3). From the other side the Combined Reactors (CR)
approach considers melt flow between simplified reactors, such as an ideal Continuous-Stirred-Tank
Reactor (CSTR or Mixer) and Plug Flow Reactor (PF). These simulations can be solved using simple fast
0
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running algorithms. However, the arbitrary chosen CR parameters will provide not adequate simulation
results. In our studies, inverse simulation was used to optimize the CR architecture and parameters, based
on CFD simulated flow pattern in the ladle, the tundish, and the continuous caster20,21).
Figure 3. Complexity of experimental and simulation methods to study steelmaking processes versus
simulation time. Arrows show inputs/outputs used in inverse simulations of Combined Reactors [20].
Methodology. The melt flow pattern in a particular metallurgical vessel can be represented by the specific
Characteristic Function. It can be the Residence-Time Distribution (RTD) for continuous flow-through
vessels, e.g. tundish or continuous caster. The RTD curve is obtained by short time tracer injection into the
in-flow stream and measuring the resulting tracer concentration in the out-flow stream. For metallurgical
processes in a closed volume (e.g., ladle), the Mixing Curve was used as Characteristic Function. The short
time tracer injection at several points located near the ladle top was done at time zero and the Mixing Curve
was obtained by tracking tracer concentration at the middle and near the bottom of the ladle.
The suggested approach20,21) includes several steps:
-Step 1: Transient CFD modeling of the melt flow in a particular design of the metallurgical vessel.
Fluent 12.0 CFD software22) was used to solve 3-D transient multiphase turbulent melt flow in: (i) a ladle
with different locations of bottom Ar-plugs, (ii) several designs of a tundish, and (iii) a continuous caster.
-Step 2: Choice of an appropriate architecture of the CR for each vessel based on the CFD results, and
-Step 3: Inverse simulation to match the Characteristic Function (RTD or Mixing Curves) obtained
from CFD and CR. In the CR architecture the several parameters, such as the reactor volumes and the flow
rates between Mixers, Plug Flow Reactor (PF), and Recirculated Volume (RV), needed to be calculated.
This was done by building a CR mass conservation spreadsheet in Microsoft Excel with calculation of the
Characteristic Function for an arbitrary set of parameters (reactor volumes and flow rates). The optimization
of these parameters in inverse simulations was done by fitting the CR Characteristic Function (CiCR) to the
“true” CFD Characteristic Function ( CiCFD). A function (φ) was minimized (Eq. 4) with the built-in Excel
Solver:
𝜑 = ∑(𝐶𝑖𝐶𝐹𝐷−𝐶𝑖
𝐶𝑅)2 → 𝑚𝑖𝑛 (4)
Ar-stirred ladle. CFD simulations were performed for a 100-ton ladle (1 m bottom radius (r), 1.2 m top
radius, and 3.4 m height) with different locations of bottom Ar-plugs (Fig. 4a): case A - one central plug
#1, case B – one plug #2 at 0.5r, and case C – two plugs #3 and #4 located on 0.5r and r apart20). To obtain
the general geometry of the melt streams and their flow rate, iso-values of vertical (Z-direction) velocity
(Vz) were plotted at different levels from the bottom at different Ar-flow rate (Fig. 4b). Based on observation
of the CFD simulated flow pattern (Fig. 4c), the architecture of the CR for the ladle was chosen as follows:
the Ar-gas driven rising plume (V0, Mixer), the top horizontal layer (V3, Mixer), the central recirculated
region (V2, Mixer), and the slow flow bottom layer (V1, Plug Flow), as shown in Fig. 4d. The values of the
independent parameters (reactor volumes and flow rate between them) were included in the inverse
simulation and varied to achieve similarity of the CFD and CR mixing curves (Fig. 4e).
(a) (b)
(c) (d) (e) Figure 4. (a) Plug locations on the ladle bottom, (b) maps of the negative (downward) melt Z-velocity
and geometry of the rising plume (empty area), (c) flow pattern (vector velocity) obtained from CFD
simulation, (d) adequate Combined Reactors (CR) architecture, and (e) fitting mixing curves at the
ladle bottom by inverse simulation of the optimal CR parameters (20 cfm Ar flow rate) [20, 21].
Once the melt flow in the ladle is described in optimized CR then it can be coupled with the thermodynamics
and kinetics of the metal-slag-gas reactions to calculate the refining processes in the ladle20). The predicted
steel de-S kinetics was in good agreement with the measured value23) and the chemistry of the melt at
different locations in the ladle can be easily simulated from the known melt flow rate between the reactors
(Fig. 5a). The calculated effect of ladle design (plug number and location) and process parameters (Ar-flow
rate) on the de-S kinetics is shown in Fig. 5b. The two-plug design is expected to give the fastest de-S
kinetics. Another interesting result is the predicted concentration differences in the different regions of the
ladle early in the process. The optimized CR approach for Ar-stirred ladle application has advantages when
compared to the other simulation methods because it can quickly provide a detailed picture of the effect of
process parameters on steel mixing, refining, and temperature distribution.
F03
F32
F20
F21
F10
V3 Mix
V2 Mix
V1 Plug
V0 Mix
V4 Slag
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ntr
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CFD ( plug 1)
CR (plug 1)
CFD (plug 2)
CR (plug 2)
CFD (two plugs 3)
CR (two plugs 3)
Case A Case B Case C
(a) (b)
Figure 5. CR simulated de-S in 100 t ladle: (a) S distribution in the ladle during refining and (b) effect of
ladle plug design on melt desulfurization kinetics [20].
Tundish. CR inverse simulation was done for the single-strand tundish of 14 metric ton liquid steel capacity
(3 m long, 1 m wide and 0.8 m melt level) with 2.6 t/min melt flow rate21). Three tundish designs were
compared (Fig. 6a): Case A - no flow control devices, Case B - with flow control devices, and Case 3 - with
a bottom Ar mixing plug (4 CFM flow rate) under the SEN. The suggested CR structure consisted of a Plug
Flow volume in-line connected to two or three Mixers/Recirculated Volume (RV) pairs. This is a reasonable
CR representation of the CFD-visualized flow pattern, and inverse simulation was used to optimize the CR
parameters by fitting RTD curves (Fig. 6b).
(a)
(b)
Figure 6. (a) Tundish designs and vector map in central vertical plane and (b) CR architecture and RTD
curves [21].
0
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%
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Bulk
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Plug 3 & 4 (bulk)
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θ
No flow control
Flow control
Ar mixing
Case A Case B Case C
Tundish design has a substantial effect on the flow pattern: poorly organized flow in the tundish without
flow control devices becomes mostly sequential flow with two easily recognized recirculation zones in the
tundish with flow control devices, and a vertical plume generated by Ar bubbles near the SEN with a poorly
organized flow pattern in the other parts of the tundish volume. Using inverse simulations, the tundish
design can be optimized based on formulated requirements, which could be a desired shape of the RTD
curve or the rate of inclusion removal.
Continuous caster. The SEN design, casting mold geometry, and casting speed all have a significant
influence on steel quality through their effect on the flow pattern in the continuous caster. Melt flow pattern
is related to the occurrence of surface defects, slag entrainment, and other steel quality problems24). The
inverse simulation was used in an integrated CFD - CR approach to describe the melt flow in the mold for
a 1500 mm × 225 mm slab20). Modeling was done for ½-slab domain and square outports located on a
central symmetry plane (Fig. 7a). A combination of vertical (Vz) and horizontal (Vx) vector velocity
components (Vz/Vx = -0.2/1) were used to simulate the downwards velocity direction at the entrance outport
(Case A). The effect of Ar-injection in the SEN was investigated in Case B. The mold wall boundary
conditions included downward translational movement at 1.5 m/min casting speed. Coupling heat transfer
with turbulent melt flow was used to establish a flow pattern and the position of the iso-thermal surface
where dendrite coherency (DC) in the solidifying steel occurs. The dendrite coherency surface was chosen
for determination of the geometry of the liquid pool. The tracer was injected through the SEN and the
Characteristic Function for inverse simulation was the RTD curve detected at the dendrite coherency
surface.
Two specific melt rotation regions (upper and bottom rolls) can be clearly identified in both cases. Ar-
injection changed the flow patterns and transformed the shape of the RTD curve. Ar-injection also had a
large effect on the structure of the both regions, decreasing the depth of the incoming jet penetration and
raising it up to the edge of the mold. The suggested CR architecture has two Mixers and one RV. The inverse
simulated RTD curves (dashed curves) vs CFD simulation (solid lines) for both cases are shown in Fig. 7b.
SEN design had an effect on the shape and the volume of the liquid pool: the conventional SEN design with
downward outports (case A) has a large volume of Mixer 1 and the RV was only about 10% of the total
volume of the liquid pool. Ar-mixing significantly increase the RV and intensified the melt exchange
between the RV and the Mixers.
For the continuous casting process, the suggested approach can be used together with traditional post-
processing CFD analysis such as melt flow instability, turbulence, and meniscus surface geometry. There
are several possible practical applications of the integrated CFD-CR approach, such as qualitative
assessment of the effect of mold design on characteristic melt flow regions and dynamic prediction of
steel composition during grade transition (Fig. 7c)20,25).
The inverse simulation played a key important role in adequate representation of the melt flow in the
described metallurgical vessels (ladle, tundish, and continuous caster) and was used to achieve similarity
between the CFD and the Combined Reactors models. A link to the thermodynamic databases will allow
the investigator to simulate melt refining and can be used as an on-line fast-running algorithm for the
entire steelmaking process control.
(a)
(a) (c)
Figure 7. (a) CFD simulated vector velocity in vertical section of continuous casting mold, (b) CFD and
inverse optimized for Combined Reactor RTD curves at dendrite coherency iso-surface, and (c) predicted
effect of casting speed on the transitional length of strand (arrows) [20].
5. 2D-3D Particle Size Conversion
Problem statement. Knowledge of the real three-dimensional geometrical topology and chemical
composition of phases and non-metallic inclusions in alloys is important for advanced analysis of
metallurgical process and product property predictions. There are two possible ways to obtain the real three-
dimensional distribution of phases: (i) using ‘true’ three-dimensional instrumental methods and procedures,
or (ii) converting two-dimensional experimental statistics, obtained from a random section, into the real
three-dimensional data (Fig. 8). The true methods such as direct extraction or in-situ observation are used
only for research purposes because they are time-consuming and costly. In comparison to three-dimensional
techniques, an automated SEM/EDX analysis provides the precise morphological and chemistry statistics
of phases, porosity, and non-metallic inclusions using two-dimensional observations of polished random
sections26). The use of inverse simulation was suggested to convert two-dimensional statistics into three-