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

in metallurgy than provide detailed mathematical analysis of this method. For those with particular interest

in general inverse simulation methodology, more information is available1-3). The paper is structured as

follows: each example has problem statement, used methodology, and illustrations of the results. In

conclusion, the described approaches were ’mapped’ together with the others used simulation and

experimental methods.

1. Thermal properties of solidified alloys

Problem statement. Casting simulation results are only useful to a metallurgy practice if they reflect the

real thermal properties of solidified alloys. A solidification path in multi-components industrial alloys could

be initially predicted by thermodynamic simulations assuming an equilibrium model (infinity diffusivity of

alloying elements in solidified alloy) or a Scheil model (restricted solid state diffusion). The real

solidification path is somewhere between these two extremes and depends on a cooling rate and melt

conditions. The two common laboratory methods (differential thermal analysis (DTA) and a differential

scanning calorimetry (DSC)) are used for determination of the liquidus and solidus temperatures and

enthalpy related properties (heat capacity and latent heat). However, these methods have problems with

detection of non-equilibrium solidus in high alloyed steels. DTA and DSC experiments do not replicate

industrial liquid treatment processes (non-metallic inclusions, gases, potential heterogeneous nucleation

sites) due used small re-melted specimen (<1 g).

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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Te

mp

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Time, sec

Texp

TmodeledCasting

Ceramic shell

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0.5

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1.5

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2.5

3

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/mK

Cp, J

/kg

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Temperature, 0C

Cp (Laser Flash)

Cp (Inverse)

K (Laser Flash)

K (Inverse)

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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Rela

tive

co

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ntr

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Time, sec

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].

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0.005

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[S],

wt.

%

Time, sec

Top

Middle

Bottom

Bulk

Measured

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[S],

wt.

%

Time, sec

Plug 1 (bottom)

Plugs 3 & 4 (bottom)

Plug 1 (bulk)

Plug 3 & 4 (bulk)

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1.6

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C

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-

dimensional volume particle diameter distribution, assuming spherical particles27,28).

0

0.2

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1

1.2

0 0.5 1 1.5 2 2.5

Ci

θ

Case A (CFD)

Case A (CR)

Case B (CFD)

Case B (CR)

0

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0 2 4 6 8 10

C

Length, m

DC (0.6 m/min)

RV (0.6 m/min)

DC (1.5 m/min)

RV (1.5 m/min)

Case A Case B

Mixer 1

RV 1

Mixer 2

Figure 8. Classification of methods used for three-dimensional microstructure characterization [27].

Methodology. Automated SEM/EDX analysis was used to obtain data for classified structural features (non-

metallic inclusions and graphite nodules) in ductile iron castings (Fig. 9a). For mono-size spheres, the

probability (P) of observing a circle with a radius r in a random two-dimensional slice is:

P(r1<r< r2) = 1

𝑅( 𝑅2 − 𝑟12 − 𝑅2 − 𝑟22)

(5)

It was assumed in inverse simulation that an arbitrary poly-sized volume spheres distribution (D3i) consisted

of k narrow overlapped normal distributions with fraction mi of each in the total particle population (∑mi=1).

These sphere distributions were ‘cut’ using Eq. 5 and the “visible” diameters D2i in a random section were

distributed into size classes. This virtual two-dimensional diameter distribution was chosen as a

Characteristic Function for inverse simulations. In the final step, the generated distribution of diameters D2i

for the arbitrary set of mi was compared and fitted to an experimentally measured distribution of particle

diameters D2exp by applying an inverse simulation. Because the shape of the distribution curve ψ(D3i)

depends on the selected bin size (ΔD3i), the independent from bin size Population Density Function (PDF)29)

(Eq. 6) was used in the weighted error function (φ’)30) (Eq. 7):

(PDF) = ψ(D3i/ΔD3i), mm-4 (6)

𝜑′ = ∑(𝑃𝐷𝐹𝑒𝑥𝑝

𝑖 −𝑃𝐷𝐹𝑠𝑖𝑚𝑖 )

2

𝐷𝑖𝑖 → min (7)

Graphite nodule distribution in ductile iron. Inverse simulation of three-dimensional graphite nodule

diameter distributions were used to determine the effect of cooling rate and inoculation on the volume PDF

in industrial castings30). The analysis revealed two different types of nodule size distribution in castings:

normal and abnormal consisted of two or three different graphite size nodule distributions (Fig. 9d).

Graphite nucleation and growth conditions both relate to the final three-dimensional nodule size

distributions in castings.

(a) (b)

(c) (d)

Figure 9. (a) Optical image of etched ductile iron microstructure in industrial 25 mm wall thickness

casting, (b) map of micro-features distribution obtained from automated SEM/EDX, (c) particle (non-

metallic inclusions and graphite nodules) two-dimensional diameter (D2) distributions, and (d) inverse

simulated three-dimensional particle diameter (D3) distribution functions (PDF) [30].

Non-metallic inclusions in steel. Cleanliness of high strength Mn-Al alloyed steel is a great concern

because of the intensive surface re-oxidation of the liquid steel. The reaction products can be formed by

multiple reactions with different Gibbs energy of formation and have diverse nucleation and growth

kinetics. The 3D size distribution of classified inclusions, obtained using the automated SEM/EDX ASPEX

system, showed27) that inclusion sizes inversely correlated to the Gibbs energy of formation: strong oxides

with more negative ΔG0 values had smaller inclusion sizes than sulfides and aluminum nitrides (Fig. 10).

Figure 10.Three dimensional distributions of different classes of non-metallic inclusion in high strength

steel alloyed by Al and Mn [27].

6. Reconstruction of Solidification Kinetics

Problem statement. Different simulation methods based on the classical nucleation and growth hypothesis

are used to describe the casting solidification. Only few experimental methods (in-situ X-ray radiography31)

and solidification interrupted by quenching32)) can deliver the real nucleation and growth rates in solidified

-3.5

-3.4

-3.3

-3.2

-3.1

-3

-2.9

-2.8

-2.7

-2.6

-2.5

-1.4 -1.2 -1 -0.8 -0.6 -0.4

Graphite

Inclusions

Y, mm

X, mm

0

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0 20 40 60 80 100

Pro

ba

bil

ity,

1/m

m2

D2, µm

Graphite

Inclusions

Total

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

0 10 20 30 40 50 60

PD

F,

1/m

m4

D3, µm

Graphite

Inclusions

Total

0

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0 2 4 6 8 10

Fre

qu

en

cy

3D diameter, µm

Mn-Si-Al-O

Mn-Al-O

Mn-S

Al-N

alloys and experimental data did not follow the classical models (Fig. 11). A method of structural

reconstruction of solidification kinetics in cast iron with spherical graphite based on inverse simulation was

suggested and tested for castings produced by different processes30).

Figure 11. Theoretical and experimental approaches used to study solidification kinetics in alloys [30].

Methodology. In the first step, the three-dimensional volume distribution of graphite nodules in ductile iron

was determined as described in the previous example. An automated SEM/EDX analysis was utilized to

separate non-metallic inclusions from small graphite nodules. This resulted in collected ‘clean’ distribution

of graphite nodule circles in polished casting sections, which was used to obtain a three-dimensional

diameter distribution. The delivered Volume Population Density Function (PDF) was used as a

Characteristic Function to reconstruct solidification kinetics.

The method of structural reconstruction of solidification kinetics is based on the assumption that the

graphite nodule PDF uniquely reflects both nucleation rate (n = f(τ)) and a growth velocity (v = ψ(τ)) of

graphite nodules in ductile iron, where τ is a relative solidification time. The integrator generates an

arbitrary number of new graphite nodules in the remaining melt at each solidification time step (20 was

used in this study). These nodules were allowed to grow with velocity (v). A two-step growth model was

chosen: (i) free growth in the melt with constant velocity until Dcritical = 2 µm and (ii) parabolic decline

graphite growth, controlled by carbon diffusion through the austenite shell, for D > Dcritical. The integrated

particle distribution curve (PDFint) was compared and fitted to an experimental (PDFexp) by inverse

simulation. Several constraints taken from experimental measurements (volume of graphite phase and

maximal diameter) were applied to an inverse simulation. The integrator determined the time-dependent

rates of nucleation and nodule growth in a virtual ductile iron, which will ‘solidify’ with a structure similar

to that which was experimentally observed in a particular casting volume.

Solidification kinetics in ductile iron. The effect of melt inoculation by FeSi additions with active elements

(Ba, Ca) on reconstructed solidification kinetics was studied in laboratory heats with pouring vertical plates

of 15 mm wall thickness30). Inoculation had a large effect on the graphite nodule number per unit of volume

(5199 1/mm3 vs 1267 1/mm3) and the shape of the PDF curve. The non-modified melt had a bell-type PDF

curve while the modified PDF curve was bi-modal (Fig. 12a). The reconstructed solidification kinetics

indicated that inoculation promoted formation of a large second nucleation wave (Fig. 12b). It was shown

that the real ductile iron solidification kinetics significantly differ from those predicted by basic nucleation

models. Cooling rate and inoculation have large effects on the kinetics of graphite nodule nuclei formation

during solidification. It was proven that the observed bi-modal PDF function in inoculated SGI is related

to the second nucleation wave. This method provides insight into the nucleation process in SGI castings

and can be used as a tool for designing effective inoculant and process control. Knowledge of nucleation

parameters can also improve the simulation of casting solidification.

Experimental

Indire

ct

(Th

erm

al a

na

lysis

)

Dire

ct

in-s

itu

mic

ro-o

bse

rva

tio

n

Macro

-mo

de

l o

f ca

stin

g s

olid

ific

atio

n

Mic

ro-m

od

els

of

eu

tectic c

ell

Theoretical

Ato

mis

tic

mo

de

ls o

f n

ucle

atio

n

Structural reconstruction of solidification kinetics

Po

st

pro

ce

ssin

g (

mic

ro-s

tructu

re)

Ato

mis

tic m

ode

ls o

f cry

sta

l g

row

th

Inte

rru

pte

d s

olid

ific

atio

n

(a) (b)

Figure 12. 15 mm wall thickness plate cast from non-inoculated and inoculated ductile irons: (a) PDF of

graphite nodule and (b) the reconstructed nucleation rate [30].

7. Discussion and Conclusions

Inverse simulation in metallurgical research is not a unique standalone tool. This method works in

combination with other experimental and direct simulation methods. That is why there is no one unified

rule which states how to use inverse simulation in particular metallurgical research. The solution of inverse

problem of IHTC is usually very sensitive to measurement error of the input data. A small measured error

would result in a large error of the calculated heat flux. The modern measurement tools provide a high

precision of measured temperature; however, error could be associated with location of the thermocouple

in the wall with high temperature gradient. Therefore, the special inverse simulation algorithms have been

developed to overcome the difficulties of the inverse problem calculation. The accuracy of these algorithms

for inverse heat conduction problem in metallurgy is discussed elsewhere14).

In this article, the several approached were described and these approaches were ’mapped’ together with

the others simulation and experimental methods for each studied case (Fig. 3, 8, 11). In each described case,

the Characteristic Functions were determined; these play a key important role in inverse optimization. Table

1 summarizes the solved problems, used approaches and Characteristic Function for described cases.

Table 1. List of the described inverse simulated metallurgical processes, used approaches and

Characteristic Function

Simulated process Model for inverse

simulation

Joint

method

Characteristic

Function

Optimization

method

Transient thermal

properties

CFD, Magmasoft Experimental Cooling curves

(casting and mold)

Riemann and

gradient errors

Ladle steel refining Combined Reactors

(Excel)

CFD (Fluent) Mixing curve (ladle

bottom)

Least squared

Tundish Combined Reactors

(Excel)

CFD (Fluent) RTD (tundish exit) Least squared

Continuous caster Combined Reactors

(Excel)

CFD (Fluent) RTD (iso-temperature

surface of dendrite

coherency)

Least squared

2D-3D particle size

conversion

Set of normal

distributions

(Excel)

Experimental Two-dimensional

nodule diameter

distribution

Weighted least

squared

Reconstruction of

solidification

kinetics

Nucleation and

growth kinetics

(Excel)

Experimental Volume Particle

Distribution Function

(PDF)

Weighted least

squared

1.E+03

1.E+04

1.E+05

1.E+06

0 20 40 60 80 100

PD

F,

1/m

m4

D3, µm

Not inoculated

Inoculated

10

100

1000

10000

100000

0 0.2 0.4 0.6 0.8 1

dn

/dt

Solidification time (relative)

Not inoculated

Inoculated

Acknowledgement

I would also like to show my gratitude to co-authors Dr. David Robertson and Dr. Mingzhi Xu of the

published articles described in this review. Great thank you Dr. Mark Schlesinger for technical comments.

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08/25/2018