computer support of casting and solidification...

COMPUTER SUPPORT OF

CASTING AND SOLIDIFICATION

PROCESS Study Support

Jaroslav Beňo

Nikol Špirutová

Ostrava 2014

Description: COMPUTER SUPPORT OF CASTING AND SOLIDIFICATION PROCESS

Author: Jaroslav Beňo, Nikol Špirutová

Edition: first, 2014

Pages: 92

Academic materials for the Metallurgy engineering study programme at the Faculty of

Metallurgy and Materials Engineering.

Proofreading: none.

Project designation:

Operation Programme of Education towards Competitive Strength

Description: ModIn - Modular innovation of bachelor and subsequent master programmes at

the Faculty of Metallurgy and Materials Engineering of VŠB - TU Ostrava

Ref. No.: CZ.1.07/2.2.00/28.0304

Realisation: VŠB – Technical University of Ostrava

© Jaroslav Beňo, Nikol Špirutová

© VŠB – Technical University of Ostrava

Content

CONTENT

COMPUTER SUPPORT OF CASTING AND SOLIDIFICATION

PROCESS .................................................................................................... 1

1. THEORY OF SOLIDIFICATION AND COOLING OF

CASTINGS .................................................................................................. 2

1.1 Solidification (crystallisation) of castings ................................................................. 2

1.1.1 Differences in the structure of the melt and solid phases ................................... 3

1.1.2 Thermodynamics of crystallisation .................................................................... 3

1.1.3 Kinetics of crystallisation ................................................................................... 5

1.2 Nuclei formation - Nucleation ................................................................................... 7

1.2.1 Homogenous nucleation ..................................................................................... 7

1.2.2 Heterogeneous nucleation ................................................................................. 10

1.2.3 Crystallisation of alloys in real conditions ....................................................... 11

1.2.4 Crystal growth .................................................................................................. 13

1.2.5 Primary crystallisation of castings .................................................................... 13

1.2.6 Dendritic growth of castings ............................................................................. 14

1.3 Solidification of castings ......................................................................................... 16

1.3.1 1.3.1 Morphology of solidification ................................................................... 16

1.3.2 Kinetics of solidification .................................................................................. 17

1.3.3 The course and duration of solidification of castings ....................................... 17

2 COMPUTER AIDED SIMULATION OF POURING AND

SOLIDIFICATION OF CASTINGS ....................................................... 21

2.1 The current situation of computer support in the foundry technology .................... 21

2.2 Options of simulation programs .............................................................................. 22

2.3 Trends in the development of simulation programs ................................................ 24

3 MODELLING AND SIMULATION ........................................................ 25

4 MODELLING OF FOUNDRY PROCESSES ......................................... 29

4.1 Classification of models .......................................................................................... 30

4.2 Physical modelling of foundry processes ................................................................ 33

4.2.1 Similarity of systems ........................................................................................ 33

4.2.2 Physical model equation ................................................................................... 35

4.2.3 Dimensionless parameters ................................................................................ 42

4.2.4 Establishment of criteria of similarity using a dimensional analysis ............... 43

Content

4.2.5 Establishment of similarity criteria using the similarity transformation method

44

4.2.6 Establishment of criteria of similarity using the method of dimensional analysis

of equations .................................................................................................................. 45

4.2.7 Overview of the most widely used dimensionless criteria ............................... 46

4.3 Mathematical modelling of Foundry Processes ....................................................... 48

4.3.1 Analytical methods ........................................................................................... 51

4.3.2 Initial and boundary conditions ........................................................................ 52

4.3.3 Numerical methods ........................................................................................... 55

5 NUMERICAL SIMULATION .................................................................. 66

5.1 Architecture of simulation programs ....................................................................... 66

5.1.1 Preprocessing .................................................................................................... 68

5.1.2 Mainprocessing ................................................................................................. 69

5.1.3 Postprocessing .................................................................................................. 69

6 THE USE OF SIMULATION PROGRAMS FOR DIFFERENT

METHODS OF CASTING ...................................................................... 70

6.1 Gravity casting ......................................................................................................... 70

6.2 Shell casting ............................................................................................................. 73

6.3 Pressure die casting .................................................................................................. 76

7 SIMULATION PROGRAMS IN FOUNDRIES ...................................... 80

7.1 Historical development ............................................................................................ 80

7.2 Overview of simulation programs ........................................................................... 81

7.3 MAGMASOFT® ..................................................................................................... 83

7.4 ProCast ..................................................................................................................... 87

7.5 PAM CAST / SIMULOR ........................................................................................ 88

7.6 WINCast /SIMTEC ................................................................................................. 88

7.7 Nova Flow & Solid .................................................................................................. 88

8 KEY TO SOLUTIONS ............................................................................... 90

Preface

1

PREFACE

Study support to the subject COMPUTER SUPPORT OF CASTING AND

SOLIDIFICATION PROCESS is primarily intended for students of combined studies. In the

combined form of study, there is a much smaller proportion of direct contact teaching, which

makes this study much more difficult for students. Our study support to the subject

COMPUTER SUPPORT OF CASTING AND SOLIDIFICATION PROCESS can help you

to eliminate this handicap to a certain extent. These is not a new textbook, there are enough of

those. The study support is a replacement, if possible, for the lack of direct instruction and

interlink to a subsequent study of professional literature itself, whether it be textbooks or

other publications.

When writing this support I have tried to maximize the clarity of the text. This cannot

be achieved, at least in my opinion, without some simplification, reduction and sometimes

even inaccuracy. If someone feels that simplification is too much, I apologize in advance. But

my experiences in teaching this subject led me to a result which is just this text.

Despite careful checking of the text, it is almost certain that I did not avoid errors,

typing errors, etc., maybe even factual mistakes. I will be most grateful, if you notify me

about them so I can gradually correct them. You can notify me either directly during

classwork or by email to [email protected] .

I wish all of you who will use this study support, a lot of strength in your study!

The Authors

Solidification and cooling of castings

2

1. Theory of solidification and cooling of castings

1.1 Solidification (crystallisation) of castings

Study time: 9 hours

Objective After reading this section, you will be able to:

define the structure of the solid phase and the melt;

define mechanisms and principles of nucleation and crystal growth

define the influence of the material properties of the mould on the

thermo-physical properties

describe thermodynamic and kinetic conditions of crystallisation; describe the basic differences between homogeneous and heterogeneous

nucleation,

describe crystallisation in real conditions

solve the critical size of nuclei; solve the time of casting solidification

Presentation

Production of castings in foundry moulds is a complex process associated with spatial

transfer, but also material transfer, with physical-chemical processes going on simultaneously

in non-stationary conditions. With regard to the time over which the process takes place, the

whole process of heat transfer between the casting and the mould can be broken down into the

solidification and the cooling of the casting.

The mechanism of solidification (crystallisation) of alloys determines the

microstructure of the alloy and therefore its mechanical properties. Solidification of alloys has

two stages:

nucleation of crystals

crystal growth

During nucleation, nuclei of future crystals are formed in many places of the melt.

Each crystal gradually grows until the crystals meet. From each nucleus, one grain of the solid

solution is formed having its own orientation of the crystal structure or particulates of another

phase (Fig. 1.). For this reason, metals are generally polycrystalline materials.

Solidification and cooling of castings

3

Fig.1. Nucleation and growth of crystals in the metal (l - melt, s - solid phase)

1.1.1 Differences in the structure of the melt and solid phases

Due to the influence of binding forces, metals and alloys in the solid state are

characterised by the regular arrangement of atoms in space, forming a crystalline lattice. Its

geometrical properties are characterised by unit cells. Each lattice, including pure metals,

includes various disorders. In the lattice, some nodal positions are not occupied by atoms, a

phenomenon called vacant sites whose number varies with temperature.

In addition to vacant sites, which belong to point defects, also line defects

(dislocations) occur in the crystal lattice which penetrate the crystal lattice along certain

planes.

Melts, the same as solid phases, belong to the condensed phases where sufficiently

close approximation of atoms results in significant distortions of the electron shell and a kind

of collectivisation of the outer electron shells, which together forms binding forces whose

effects maintain atoms in certain positions.

The best known observable difference between the liquid and solid states is the

fluidity, i.e. the ability to fill the space in which the substance is located. Significant

differences between the states considered include greater entropy of the liquid, greater

compressibility and thermal expansion of the melt, and higher diffusion coefficient in the

melt.

According to the "Vacancy Theory", or the Theory of Holes, developed by Eyring, or

Frenkel, the spatial arrangement of atoms in a melt is similar to the solid state arrangement

except the significantly higher number of vacancies. The difference between the solid phase

and the melt is primarily reflected in the density of vacancies which is in the melt near the

melting point significantly higher (by several orders of magnitude) than in the solid phase

near to this temperature. In other words, the number of holes increases with the temperature,

the same as the total volume of the melt. And vice versa, a transfer from the liquid phase to

the solid phase (in line with the change of the arrangement from near to greater distance)

results in volume changes.

1.1.2 Thermodynamics of crystallisation

Crystallisation is a phase transition accompanied by a change in volume, releasing

latent heat of crystallisation. The crystallisation process is governed by general rules

applicable to all phase transformations.

A general criteria used to assess the possibility of spontaneous process of any phase

transformation is a change in free enthalpy G, defined by:

STHG

Thermodynamics and kinetics of crystallisation

4

Where H is the change in enthalpy, and S is the change in entropy accompanying

this transformation. Free enthalpy is the proportion of energy which the system can transform

into work, i.e. in this case, to realize a phase transition.

During crystallisation of pure metal, a change in free enthalpy is given by the

difference in enthalpies of the metal in the solid (GS) and liquid (GL) states:

LS GGG

There is a temperature at which the initial phase and the emerging phase is in a state of

thermodynamic equilibrium. The condition for the equilibrium is the equality of the free

enthalpy DG of both phases (Fig. 2).

Fig.2. Change of the free enthalpy G with the temperature, for the solid and liquid phase

of metal

Crystallisation is caused by the metal or alloy trying to achieve a stable state when

being cooled. From the thermodynamic laws' point of view, a stable state is defined by a

minimum of free enthalpy.

Above the temperature T0 (Figure 2), the values of free enthalpy of the melt are lower

and therefore it is thermodynamically more stable than the crystalline phase. At temperature

T0, there is equilibrium between the two phases. This is referred to as the theoretical

crystallisation temperature.

Since crystallisation is a diffusion process, it can not be expected to begin at T0. There

must be some supercooling T below this temperature, so that the change in free enthalpy G

between the original and the newly formed phases will be sufficient to induce crystallisation.

The difference in the enthalpies (G) provides all the work necessary for the formation and

growth of nuclei.

In practice, actual formation of the nuclei of crystallisation occurs with an

undercooling of less than 10 °C.


5

1.1.3 Kinetics of crystallisation

In terms of kinetics, crystallisation consists of two independent processes, i.e.

nucleation and crystal growth, depending on the temperature gradient (planar, cellular,

dendritic). In other words, a transfer of the interface towards the liquid phase. Each of these

major processes includes several other sub-processes. Where the slowest process limits the

speed of the entire process. What happens in the case of nucleation, for example, is

accumulation of the appropriate types of atoms through diffuse or other movement, their

internal organisation, formation of interphase boundaries, etc. Similarly, the growth involves

transport of atoms through the old stage, their jump across the interphase boundary and the

transport of atoms through the new phase. Most of these processes are thermally activated, i.e.

the energy barriers are overcome by thermal motion of energy activated atoms or their groups.

Transition of the melt into a solid crystalline phase can be studied from two

perspectives:

In terms of growth rate of the new phase, depending on the conditions of heat

transfer from the melt. In other words, based on the volume of the solid phase

grown under the cooling conditions.

From the physico-chemical point of view (mechanism of solidification)

High speed of nucleation (a large number of nuclei) and a low speed of further crystal

growth results in a structure composed of fine grains and, vice versa, low speed of forming

crystallisation centres and high linear speed of growth results in large polyhedral grains.

According to Tammann, the rate of nucleation KZ is defined by:

𝐾𝑍 =𝑐𝑜𝑢𝑛𝑡 𝑜𝑓 𝑛𝑢𝑐𝑙𝑒𝑢𝑠

𝑐𝑚3 ∙ 𝑚𝑖𝑛

and the speed of dendritic growth KG:

𝐾𝐺 =𝜆 ∙ (𝑇𝑝𝑜𝑢𝑟𝑖𝑛𝑔 − 𝑇𝑠𝑜𝑙𝑖𝑑)

𝐿 ∙ 𝑥

where: - thermal conductivity of the liquid metal

L - latent heat of the crystallisation

x - thickness of the liquid phase layer

Graphic dependence of linear growth changes depending on the degree of

supercooling is shown in Fig. 3a and 3b.


6

3a Change in linear growth rate depending

on the degree of supercooling

3b Change in linear growth rate depending

on the degree of supercooling (metals and

alloys)

Fig.3. Change in linear growth rate

Therefore, a finer microstructure occurs when the cooling rate of the melt is increased,

thus achieving a higher driving force of solidification. Fine grain materials and materials with

fine particles of the phases have better strength properties compared to coarse-grain ones.

This procedure, however, can only be used in small volumes of melt because in large volumes

of melt the cooling rate cannot be increased too much as it is limited by heat removal from the

molten alloy.

For this reason, inoculation is often used to make the structure finer. The principle of

inoculation is a deliberate introduction of extraneous fine particles in the melt which become

crystallisation nuclei, resulting in a fine microstructure as well.

The crystallisation nuclei can occur spontaneously (homogeneous nucleation) or are

introduced into the melt (heterogeneous nucleation).

Homogenous and heterogeneous nucleation

7

1.2 Nuclei formation - Nucleation

The term nucleation refers to the formation of a new phase which in the case of

crystallisation is separated from its surroundings by a discrete boundary.

Spontaneous nucleation, without the effects of walls, inclusions, additives or pressure

pulses, occurs very rarely, in general, we can say it is only nucleation under laboratory

conditions. In such cases, we speak about homogeneous nucleation. In actual conditions,

nucleation usually begins on the surface of the mould, coquille or different particles present in

the melt. In this case, we speak about heterogeneous nucleation.

1.2.1 Homogenous nucleation

This term refers to the formation of nuclei of a new phase within the area of the old

phase. It arises from the centres of clusters of atoms, which are formed in the melt naturally.

When the temperature of the melt approaches the solidification temperature (Tt), it leads to a

natural fluctuation through concentration, which creates clusters with geometrically regular

arrangement of atoms which corresponds to a crystalline state of mass, in other words to the

formation of a new phase. The amount nuclei formed per unit of time and per unit volume can

be expressed as a statistical probability of homeo phase fluctuations.

Changes in the concentration are also accompanied by changes in the energy system,

i.e. thermal fluctuations. These agglomerates become actual (active) nuclei once they are

thermodynamically stable. This means that they have less free energy than the original liquid

phase.

The overall change in the free enthalpy during the formation of nuclei of the solid

phase in the melt is determined by releasing the free enthalpy during the formation spherical

nuclei with a radius r and energy necessary to create the surface of the nuclei (nucleus - melt

boundary). Because of the low volume of nuclei, and therefore a large surface to volume ratio,

the interface energy plays an important role.

The work required to create a nucleus is proportional to the free enthalpy which for the

formation of a nucleus in the melt is given by:

SV GGG

where: G - total change in free enthalpy

GV - change in free enthalpy of the system during the transition of the liquid

phase in the solid phase

GS - free enthalpy required to create the interface

When creating a spherical cluster of the size r, a certain amount of energy must be

released. This quantity is defined by the product of the volume of the cluster (

3

3

4rV

) and

free energy (free enthalpy) of the volume unit:

m

mV

V

GrGE

3

13

4

where: Gm - difference of molar enthalpies of the solid and liquid phases, which is

negative during crystallisation


8

Vm - specific molar volume

The value of free energy GV, by which the energy of the system will be reduced,

takes on a negative value (with respect to the system) and makes the system

thermodynamically stable.

The value of the energy required to create the crystal - melt interface is proportional to

the size of the area (A) of the phase boundary () and the magnitude of the surface tension.

This energy represents the increase in energy of the system, therefore it is positive. The

formation of an interface for a spherical nucleus is therefore given by the equation:

2

2 4 rGE S

The total free energy (enthalpy) during the formation of crystallisation nuclei in the

melt is defined by:

m

mSVC

V

GrrGGGE

32

3

44

Energy relations during the formation of nuclei can also be described graphically (Fig.

4), the waveform E2 corresponds to a quadratic parabola, curve E1 has the character of a cubic

parabola.

Fig.4. Change in the free enthalpy (G) of a nucleus depending on its radius

The maximum value of the cumulative curve EC = E2 - E1 is reached at the critical size

of the nucleus rkr. Due to the critical size of the nucleus, the particles which are smaller than

rkr will dissolve back, while particles larger than rkr further increase because both processes

are associated with a decrease in the free enthalpy.

The critical size of a nucleus can be determined using the first derivative to determine

the overall free energy (setting of the extreme of function) to be equal to 0 and subject to the

condition:

0

r

EC

0

r

EC


9

then the critical nucleus size can be defined according to the equation:

T

MT

G

Vr

m

mkr

lg

22

where T - the actual crystallisation temperature

M - molecular weight of the crystallising substance

The work associated with the formation of a nucleus may be derived from the equation

m

mkr

G

VE

3

3

16

The critical size of a nucleus rkr is inversely proportional to the value of supercooling

T (Fig. 5). With an increasing value of supercooling T the radius of the critical size of the

nucleus decreases. Critical supercooling of the melt, when a homogeneous nucleus becomes

stable and is capable of further growth, is by DAVIES (Fig. 6.):

tkr TT 2,0

Fig.5. Dependence of the critical nucleus

size on supercooling

Fig.6. Effect of supercooling on the course

of nucleation


10

1.2.2 Heterogeneous nucleation

In reality, the heterogeneous nucleation takes place at moderate supercooling.

Crystallisation is induced by the presence of various inclusions - oxides, silicates, nitrides,

and non-metallic inclusions, etc., it also occurs on the walls of the mould and uneven portions

of the mould. This is a common crystallisation, without any need for the necessary

supercooling to initiate spontaneous crystallisation. For crystals to form, supercooling of

about 0.02 Tt is sufficient. This means that this crystallisation precedes the spontaneous

crystallisation. Heterogeneous nucleation occurs much faster than the homogeneous

nucleation

For an inclusion to become a foreign crystallisation nucleus, it must meet certain

criteria:

it must have a related crystalline lattice

the nucleus must be wetted by the melt; The greater the affinity of the

crystalline lattices of the metal and the inclusions, the smaller the wetting

angle, and the easier it becomes for a nucleus to become a crystalline nucleus

inclusions purposefully brought into the melt, during crystallisation act as

active pads; these are called inoculants (e.g. Ti and Zr in Al alloys and FeSi in

cast iron).

But there are other factors that affect the actual course of crystallisation:

Purity of the metal (if the metal contains gases, inclusions, air, then it does not

need substantial supercooling to initiate crystallisation)

Movement of the melt (any movement of the melt, such as vibration, flow, etc.

reduces the supercooling necessary for the formation of nuclei)

Pressure (high external pressure helps crystallisation)

Level of overheating of the metal; at the lowest level, the structure is the

finest, the size of the grain increases with increasing level of overheating and

at a certain critical temperature the structure becomes finer again

Physical basis of crystallisation from heterogeneous nuclei consists in reducing the

interface tension in the melt - foreign particles - resulting nucleus, therefore the value of the

energy required for the formation of an active nucleus is lower. For this reason, the

heterogeneous crystallisation occurs already with low supercooling.

Interface surface tension and wettability between the nucleus and the melt are the most

important variables which determine whether a foreign nucleus becomes active. The surface

and interface tension and the wetting angle between the melt (T), the nucleus (Z) and the

crystalline phase (K) arising from a foreign nucleus (Fig. 7) can be defined by the equation:

cos KTKZTZ

where the wetting angle is:

KT

KZTZ

cos


11

Fig.7. Heterogeneous nucleation

Crystallisation from the melt of the given nucleus is therefore dependent on good

wetting on condition that:

KZTZ

The smaller the wetting angle the greater is the assumption that the nucleus will be

active, capable of faster growth and thermodynamically stable. The greater the affinity of the

crystalline lattices of the metal and the inclusions, the smaller the wetting angle, and the

easier it becomes for a nucleus to become a crystalline nucleus. The energy required for the

formation of a heterogeneous nucleus is given by

2

33

..3

coscos324

V

KTzh

GG

And the critical magnitude of the nucleus dimension is

2

2

V

KTkr

Gr

1.2.3 Crystallisation of alloys in real conditions

Crystallisation conditions in the process of solidification of alloys can be assessed

using equilibrium phase diagrams (Figure 8.). However, their validity is limited to the

processes of cooling of alloys at very low cooling rates. Due to the existence of a temperature

difference between the liquidus and solidus, at each temperature solid and liquid phases in

thermodynamic equilibrium are given with different chemical compositions.

Fig.8. Crystallisation of alloys in real conditions

Crystallisation of alloys in real conditions

12

The composition I corresponds to concentration C0. When reaching the temperature of

liquidus (tl), crystals of the melt with the composition Ck become solid first. By precipitation

of crystals of the solid phase, the composition of the melt is gradually changed along the

liquidus curve, so the rest of the melt solidifies at the temperature (ts) with the concentration

CL. Under equilibrium conditions, the concentration of the melt will change from C0 to CL and

the concentration of the solid phase from Ck to C0. The ratio of concentrations of additives in

the crystals and the melt can be characterized as "partition coefficient K". This coefficient is

defined by equations:

10

C

CK K

S

10 L

LC

CK

10

LSK KK

C

CK

Gradual change in the concentration of the solid phase leads to inhomogeneities (first

crystals rich in component A, up to the last solidifying ones depleted of component A). In this

way an overall heterogeneity of crystals is created, which is partly equalised by the diffusion

in the solid and liquid phases, but not completely.

After the end of crystallisation, all concentration differences should be equalised. With

rapid cooling of casting, however, the segregation will remain preserved.

In real conditions of crystallisation, only a very low efficiency of diffusion is to be

expected. In spite of the higher mobility of atoms in the melt a uniform concentration of each

element in the close vicinity of the crystal cannot be expected. Changing the concentration

CL(x) of the additive element in the melt (Fig. 9) in the vicinity of a growing crystal causes a

change in the liquidus temperature (tL(x)) which is lower in the vicinity of the interface and in

the direction into the melt it increases to the equilibrium liquidus temperature tL.

Fig.9. Changing the concentration of the additive element B in the melt at the distance of

the crystallisation queue.

The lowest temperature (highest supercooling) is achieved at the mould-metal

boundary, towards the centre of the casting the temperature increases and the maximum value

is reached in the heat axis of the casting, i.e. in the place which solidifies last (Fig.10)


13

Fig.10. A curve of the temperature and constitutional supercooling of the melt

The difference between the progress of the actual temperature (tt) and the change in

liquidus temperature (tl(x)) is the so-called constitutional supercooling (tk) whose value

increases with increasing distance from the crystal boundary and then falls to zero.

Constitutional supercooling affects the resulting primary structure of the crystallising melt

and is the cause of branching in the growth of metal crystals.

1.2.4 Crystal growth

For crystals to grow once nuclei have been formed, greater heat flux is required from

the casting than from the centre to the surface, it occurs from thermodynamically stable

(active) nuclei of crystallisation while the free energy G of the system decreases. In other

words, permanent removal of latent heat of solidification from the interface. This is only

possible at a certain temperature gradient in the area adjacent to the solid phase - liquid phase

boundary.

Generally, the growth rate is an exponential function of the energy conditions of the

growth and temperature. The basic question of the growth mechanism is the way of

attachment of atoms on the surface of the growing nuclei. To capture atoms from the melt,

appropriate levels must exist on the surface. At first, crystal growth occurs from individual

nuclei (micro scale), later a continuous layer grows against the direction of heat removal over

a given time (macro-scale).

Under these conditions, crystallisation does not follow the equilibrium conditions. It

would be achieved only at low rate of cooling. A branched crystallisation structure can be

divided into several structures: a planar structure, cellular structure, cell-dendritic and

dendritic structure.

1.2.5 Primary crystallisation of castings

For the internal (exogenous) nuclei of crystallisation, there is a basic rule saying that

when they are in contact with the walls of the mould, they preferably grow into stable


14

crystals; then the cast primary structure on the casting surface must consist of that many

crystals growing perpendicular to the wall surface of the mould.

In real conditions (Fig. 11), the surface area of a casting contains randomly oriented

crystals (crystallites). Due to the rapid cooling, this casting crust has different mechanical

properties than the centre of the casting and the growth of the nuclei which are in good

contact with the walls of the mould is preferred. Local preferential growth occurs in areas of

increased thermal conductivity.

This casting structure is linked with a region of elongated columnar crystals whose

main axes are parallel to the direction of maximum heat removal from the casting and which

have a typical dendritic characteristic.

In the centre of the casting, there is a region of equiaxed globular (polyhedral) crystals.

These types of structure may not always be found in castings. The structure of castings is only

comprised of a columnar crystals which meet in the thermal axis (trans crystallisation), or

vice versa, the entire structure is equiaxed. This refinement can be achieved by other

interventions to solidification, such as inoculation or crystallisation disturbed by external

forces (vibration, ultrasound, etc.)

Fig.11. The most frequent structure in castings

In technical alloys, primary crystallisation is followed by phase transformations in the

solid state, the so-called recrystallisation, which also affect the final structure of the casting

(e.g., ferrite transformed in austenite when temperature decreases).

1.2.6 Dendritic growth of castings

This type of crystallisation is typical for Fe alloys. Dendritic growth is based on high

crystallisation rate. The conditions of high crystallisation rate and supercooled layer of the

melt before the interface (constitutional supercooling) lead to a significant application of

crystallographic effects, energy faults and projections - inequalities on the boundary surfaces

of the nuclei (crystals). Atoms, diffusing from the melt in an upstream direction of the

cooling, start depositing on them faster and the crystals grow and elongate into multiaxial

trees (Fig. 12) with a prominent main axis.


15

Fig.12. SEM image of a dendrite and its model

With increasing rate of crystallisation, the dendritic structure develops more clearly,

forming dense network of secondary and tertiary axes in a tree-like shape. At high cooling

rate, the distances between the primary branches get smaller, and the secondary and tertiary

branches tend to disappear.

Modelling and simulation

16

1.3 Solidification of castings

Solidification is a progress of crystal layers, their directional orientation for the

purpose of high internal homogeneity of the casting and a number of solidification

phenomena accompanying solidification. The concept of solidification has a more general

meaning than the term of crystallisation. In addition to phase transformation, it involves

morphological, physical and volumetric changes. In view of the above processes, we are

mainly interested in solidification kinetics

1.3.1 1.3.1 Morphology of solidification

Based on the solidification process inside the casting, there are two types of

solidification morphology

Exogenous

Endogenous

In the exogenous solidification, nuclei are found on the surface of the mould and

solidification process advances from the surface to the centre of the casting. The types of

solidification include (Fig. 12):

a1) Solidification on a smooth layer of crystals (with a planar interface).

a2) Solidification on a rugged interface.

a3) Spongy solidification with strongly branched dendrites.

In addition, endogenous solidification is accompanied by nuclei and crystals being formed

from them across the entire volume of the melt. There are the following types of solidification

processes (Fig. 13):

b1) Mushy solidification. It is a bulk solidification.

b2) Layered solidification - also a bulk solidification; however, a layer of globular crystals is

formed from the surface.

Fig.13. Morphology of solidification


17

1.3.2 Kinetics of solidification

In reality, only solidification of alloys in a temperature interval is taken into

consideration. During solidification, three zones can coexist (Fig. 13). From the surface area

in contact with the mould there is a solid metal zone (x) whose thickness increases

continuously over time. Next to it, there is a two-phase zone () whose width depends on the

period of solidification and the temperature gradient, and it widens over time. The last zone is

the melt zone whose width is constantly decreasing.

The thermal axis is the set of points where the crystallisation surfaces meet (iso-

solids). The width of the two-phase zone, which is limited by the areas of iso-solid and iso-

liquis fractions, affects the occurrence and extent of the microporous structure. In the case of

wide zone, isolated islands of the melt are formed in the heat axis, whose solidification and

shrinkage leads to micro-shrinkage. The width of the two-phase zone is affected by:

Interval of solidification of the alloy (defined by the chemical composition of the

alloy

Rate of cooling (heat accumulation of the mould bf)

ffff cb

where: f - thermal conductivity of the mould

cf - specific heat of the mould

f - volume mass of the mould

The higher the accumulation capacity of the mould, the higher the transverse

temperature gradient, and the narrower the two-phase zone is.

1.3.3 The course and duration of solidification of castings

The process of solidification of a casting from the mould wall takes place at a certain

speed which can be assessed by the thickness of the solidified layer of alloy per unit of time

(Fig. 14.). Calculation of the solidification time is based on the thermal balance of the casting

and mould during solidification:

Q1 = Q2 [J]

Where the subscript 1 specifies the casting, the subscript 2 is the mould

For a semi-infinite plate-shaped mould, Q1 can in a simple form be expressed as:

)( 111111 sTTcLxSQ

To determine the amount of heat taken (accumulated) by the mould, we use the

equation of temperature distribution in the mould which corresponds to the course of the

Gaussian curve.

a

xGdue

TT

TTu

u

vp

v

2(

22

022

22


18

Rate of heat transfer from the metal into the mould corresponds to the heat flow

density:

a

TT

x

t

d

dQq

vp

s

)()(

22

)(

Fig.14. Conditions of solidification at the mould - metal interface

And the amount of heat passing the entire area of the casting:

f

vp

fff

vps

bTTS

cSTTdqSQ

)(2)( 221122

0

)(2

After reducing the above equation and substituting into the equation for Q1 = Q2:

f

s

bTiSTTcLV 11111 2)(

provides a definition equation for calculating the time of solidification:

Where the fraction S

VR

is the relative thickness (module) of the casting, as defined

by CHVORINOV. The relative thickness of the casting is the ratio of the volume of the

casting (its heat capacity) to the surface. Then, increasing robustness of the casting (with

increasing R) extends the time of solidification.

22

2

11

2

12

1

1

4

)()(

f

s

bTi

ttcL

S

V


19

Summary of the chapter (subchapter) concepts

Free enthalpy

Nucleus

Homogenous nucleation

Heterogeneous nucleation

Constitutional supercooling

Segregation

Dendrite

Heat accumulation of the mould

Module (relative thickness)

Questions about the studied subject

1. What is the progress of solidification of alloys, what stages are there?

2. What determines the microstructure of an alloy and therefore its mechanical properties?

3. How can the characteristics of a microstructure be affected?

4. What causes a spontaneous process of any phase transformation?

5. What causes crystallisation of metals or alloys?

6. What is the difference between homogeneous and heterogeneous nucleation?

7. Which way of nucleation occurs in real conditions of solidification of metals and alloys?

8. What conditions must apply for an inclusion to become a nucleus of crystallisation?

9. What is the physical nature of the crystallisation from heterogeneous nucleation?

10. What is segregation?

11. What types of crystals can we find in the structure of castings?

12. What kinds of solidification do we know

13. What is the heat axis

14. How can the width of a two-phase zone in casting solidification be influenced


20

References for further study

HAVLÍČEK, F.: Teorie slévárenství (výběr z přednášek). VŠB – TU Ostrava, Ostrava, 1992.

s.130

JELÍNEK, P.: Slévárenství. VŠB – TU Ostrava, Ostrava, 2000. s.251

MYSLIVEC, T.: Fyzikálně chemické základy ocelářství. SNTL, Praha, 1971, s. 445

MICHNA, Š., NOVÁ, I.: Technologie a zpracování kovových materiálů. ADIN, Prešov,

2008, s. 326, ISBN 978-80-89244-38-6

VOJTĚCH, D.: Kovové materiály. VŠCHT Praha, Praha, 2006, s. 185, ISBN ISBN: 80-7080-

600-1

PŘIBYL, J.: Tuhnutí a nálitkování odlitků. SNTL, Praha, 1954, s.312

KUBÍČEK, L.: Krystalizace kovů a slitin. VŠCHT Praha, Praha, 1991, s. 238, ISBN 80-7080-

130-1

KUCHAŘ, L. Metalurgie čistých kovů. VŠB – TU Ostrava, Ostrava, 1988, s. 338

STEFANESCU, D.M. Solidification and modeling of cast iron—A short history of the

defining moments. Materials Science and Engineering A 413–414 (2005) 322–333


21

2 Computer aided simulation of pouring and solidification of castings

2.1 The current situation of computer support in the foundry technology

Study time: 2 hours


define possibilities of utilising computer technology in foundry practice;

which processes associated with solidification and cooling of castings

can be defined and solved using simulation programs. Describe

differences between the various commonly used simulation programs

describe the main issues addressed by simulation programs in the area of

filling sand moulds, permanent moulds and the area of pressure casting;

in the phase of solidification and cooling, together with material

properties and the possibility of changes during the preparation process

for the production of castings ......

Presentation

Production of castings in foundry moulds is a complex process associated with spatial

transfer, but also material transfer, with physical-chemical processes going on simultaneously

in non-stationary conditions. With regard to the time over which the process takes place, the

whole process of heat transfer between the casting and the mould can be broken down into the

solidification and the cooling of the casting.

Without the use of computer technology, solidification and crystallisation of metals

and their alloys was most frequently assessed using metallographic analyses of

macrostructure and microstructure. In the 1980s, the first simulation software were made

focused on solidification of castings. Improvement of computer technology and the

development of experimental techniques led to improved simulation programs which can be

used to study and monitor solidification of castings, not only in the overall period, but also in

short intervals. From this perspective, the numerical simulation have become a frequently

used tool effectively exploited, not only to optimise the proposed technology of casting

production, but also a focal point of research in thermal processes in the casting-mould-

surroundings system.

Modern simulation programs include prediction of the melt during mould filling,

interaction of the metal and mould, casting deformation - stress, and prediction of the

structure and microstructure of the casting. The models include the heat transfer, rates and

methods of moulds, the flow of metal in the mould, solidification kinetics, formation of


22

structures, models, porosity, segregation in the solidification interval, and finally calculation

of the tension.

Although each technology of casting production has its own differences and specific

procedures, there are currently complete numerical simulations for the most commonly used

technologies of the production of castings. In particular, this involves technologies of gravity

casting in sand and metal moulds, as well as making models for the low- and high-pressure

die casting methods, the lost foam method, and for the methods of semi-solid processes.

2.2 Options of simulation programs

Constant development of hardware and software goes hand in hand with higher

accuracy of the calculations, and simulation programs have been constantly improved. Using

simulation programs, the following groups of issues are currently addressed:

In the phase of filling a classic sand mould:

calculation of the mould filling time using different criteria

melt filling method and places of turbulence and vortex formation,

monitoring of pressure and temperature in the melt

velocity of metal in various parts of the system (the nature of the flow depends

on the value of the Reynolds criterion)

In the stage of solidification:

solidification times, temperature gradients and cooling conditions in any point

calculation of temperature fields, proportion of the liquid phase, shrinkage and

shrinkage porosity

thermal load of cores and moulds

cooling curves in any area

efficiency of exothermic or insulating attachments

segregation of components

In the phase of cooling and related material properties:

stress distribution in the casting and components of moulds and cores

deformation of the casting and mould in relation to time and temperature distribution

thermal and diffusion flux

determining the structure of the material in different stages of cooling

calculation of the transformation time (according to ARA, IRA diagrams)

calculation of the mechanical properties of the material, calculation of hardness

inclusion of the impact of the mould on the course of graphitic expansion

For the technology of casting in metal moulds (gravity or pressure casting)

distribution of the temperature field in various stages of the production cycle

thermal load of cores and moulds

metal flow


23

strain in different parts of the mould and casting

specifying pressures for individual process steps

specifying technologically optimal times

temperature regime at the start of production

opening and closing of the mould to be defined depending on the time or the

temperature

checking the function of cooling channels

The technologist can use simulation software to modify or adjust:

optimise the inlet system

optimal location of the riser and chillers

reduction of the size and number of the riser and chillers

minimisation of residual stresses and optimisation of stress distribution after cooling

estimate and minimise distortion, warping and shrinkage

optimise the conditions of filling in the pressure die casting, times and optimisation of

the casting cycle, reduction of the thermal stress of cores

improve the function of the cooling channel based on the information from the

thermocouple in the die casting

The quality of the simulation programs, their information value including the

compliance of simulation results with real processes is primarily affected by the following

factors:

1. the quality of mathematical description of sub-processes - i.e. using the Fourier

differential equation of heat conduction which is strongly influenced by the correct

choice of initial and boundary conditions;

2. including a variation in the behaviour and condition of the cast material from the ideal

state of a single-phase melt (e.g., a non-Newtonian fluid, temperature dependence of

gradual release of latent heat during solidification of the melt, etc.);

3. thermo-physical definition of properties of the moulds and the cast material depending

on the temperature across the entire required width of the temperature range.

An equally important factor is the fact how these simulation programs define the flow

of fluid through the law of conservation of mass (continuity equation) and momentum

(Navier-Stokes law), heat transfer during solidification and cooling of the castings (Fourier

differential equation), the level of residual or internal stresses, laws of mechanics of rigid

bodies in plastic and elastic deformation, etc. This is closely associated with the choice of

initial and boundary conditions of the solution, which significantly affects the results of

numerical simulations.

Determining the values of the required thermo-physical quantities depending on the

temperature can be an issue in simulation calculations. This is the most common cause of

differences between the results obtained by simulation calculations and experimental

measurements under comparable conditions.


24

Numerical simulation and modelling plays an important role in optimizing the

planning and foundry processes. The purpose of modelling - simulation is to achieve a

forecast with the greatest possible accuracy, saving time and money in the management,

operation, development and production. Efforts to achieve a significant increase in

productivity, improve quality and accelerate the innovation process lead to the exploitation of

the results obtained from the numerical simulation of other processes. There are efforts to

incorporate simulation into information and optimisation technologies, or into other technical

calculations which can utilise the analyses performed (for example, the use of the distribution

of residual stresses in the casting in subsequent crash tests of cars). Everything leads to the

creation of a virtual test medium which will maximise the improvement of the product during

the design and prototype production phases.

2.3 Trends in the development of simulation programs

In the last few years of the manufacture of foundry castings, there have been many

significant improvements, especially regarding the possibility of the application of

computational simulation tools. The use of simulation in the casting process has a significant

effect on the suppression of shrinkage, increased use of liquid metal and optimisation of the

inlet and exhaust systems for high-pressure die casting moulds. The development of these

tools now continues in the much wider context than before. This creates new and improved

modules for more casting technologies. Companies engaged in the development and sale of

these programs put a significant amount of effort and financial resources to meet the needs of

the market and their customers. Research and development is focused on the following areas:

improvement and acceleration of numerical calculation methods

more precise and complete databases of thermo-physical data, heat transfer

coefficients and other parameters necessary for the calculation

possibilities of calculations of new foundry processes and materials

development of models for the micro-modelling

improved criteria for the evaluation of simulation results

implementation of optimisation techniques in numerical simulation

possibility of using the simulation for further technical computing, information

and control processes


Modern simulation programs include prediction of the melt during mould filling,

interaction of the metal and mould, casting deformation - stress, and prediction of the

structure and microstructure of the casting. Models include the heat transfer, rates and

methods of moulds, the flow of metal in the mould, solidification kinetics, formation of

structures, models, porosity, segregation in the solidification interval, calculating also the

tension.


1. What can be predicted in the solidification process using simulation programs

2. What is the purpose of modelling or simulation of foundry processes?


25

3 Modelling and Simulation

Study time: 1 hour


define the basic concepts of modelling and simulation used in the study

of general systems and the differences between them

define the steps of modelling and simulation processes

describe the process of simulation of a real system

Presentation

The term modelling refers to an experimental process used to gather information on a

system using a different system - model. The system means a set of elementary parts,

elements, that have mutual specific links. Since a model is also a system, this similarity is

used in modelling. The significance of modelling consists in the fact that information about a

system is obtained in a more suitable, faster and often more cost-effective way by

experimentation on the models than on the original. Generally, any system can be studied

using the following scheme (Fig. 15):

Fig.15. General principle of studying any system

The basis of modelling is a replacement of the studied system with the model (more

precisely: with a system which models it), the objective of which is to gain, through

experiments with the model, the information about the original system under investigation


26

For simple systems being modelled, the system behaviour can be defined by

mathematical relationships and the desired values determined using mathematical methods.

The results are functional relationships in which model parameters are used as variables.

A comprehensive analysis must be performed for more complex systems whose

specific properties include large extent, incomplete information, qualitative nature of

parameters, dynamic nature of progressing processes and complex nature of the relationships

between elements of the system. In such a case, modelling is generally done in several steps:

1. Creating an abstract model - formulated on the basis of purpose-built and a

simplified description of the investigated system

2. Creating a simulation model - created by entering an abstract model using a

programming language (a simulation program)

3. Simulation - experiments using the simulation model. The aim of this stage is to

analyse the behaviour of the system depending on the input values and the values of

the parameters. The process of simulation consists of repeated model solution,

performing simulation runs in which the output data defining the behaviour of the

system are evaluated. The simulation runs are repeatedly carried out until sufficient

information is obtained about the system, or until such parameter values are found for

which the system has the desired behaviour.

Simulation is a research technique based on replacing the investigated dynamic

system with its simulator, performing experiments with the simulator in order to obtain

information about the original dynamic system examined.

Before the simulation itself, the first step is carried out - verification of the simulation

model, i.e. verification of the correctness of the model. The purpose of verification of the

model is, for example, to rule out potential bugs in the program or make sure an inappropriate

numerical method is not used.

Another equally important step is a constant confrontation of information which we

have about the system being modelled and which we are to obtain through the simulation.

This leads to the verification of validity of the model. Verification of validity of the model is a

process in which we are trying to prove that we are actually working with a model adequate to

the modelled system.

If the behaviour of the model does not match the expected behaviour of the original

model, the model must be modified in the light of information obtained by previous

simulations (see Fig. 16)


27

.

Fig.16. Procedure of simulating a real system

The process of simulating a real system can be summarised in several steps:

1. determine the purpose of simulation and the monitored outputs

- the participating processes can be determined based on the outputs

2. create the simulation model

- isomorphic relationship with the abstract model

this includes filling the model with data

3. model validation

4. create a computer model

5. verify functionality of the computer model

6. design experiments

7. process the results

- record the simulation process

- visualisation, animation

- analysis, comparison with real data and selection of the best solution


System

Model

Modelling

Simulation

Model verification

Model validation


28


1. What is meant by the term system?

2. What is the basis of modelling?

3. What is simulation?

4. What does the term model verification stand for?

5. What does the term model validation stand for?

Modelling of foundry processes

29

4 Modelling of Foundry Processes

Study time: 20 hours


define methods of the modelling of casting processes

describe the criteria for classifying models, define the basic concepts of

physical and mathematical modelling

describe the differences between physical and mathematical modelling

define similarity of systems

resolve the determination of the criteria of similarity using dimensional

analysis

resolve the determination of the criteria of similarity using similarity

transformation

resolve the determination of the criteria of similarity using dimensional

analysis equations

define the basic equations of physical modelling of foundry processes

define the numerical methods used for the simulation of foundry

processes

describe analytical and numerical methods of mathematical modelling

describe and solve the conditions for uniqueness

Presentation

As mentioned in the previous chapter, modelling is an experimental process in which

the studied system (original) is matched with another system, based on given criteria, either

physical or abstract, called a model.

The aim is to capture the truest possible behaviour of the real system using a

mathematical or physical model. Based on the results achieved with the model, it is possible

to predict the behaviour of the real system under various changes to the process. The role of

modelling is to further develop the theory of physical, chemical and thermal processes, and

further use these theories in the practice of modelling.

Using modelling for industrial equipment, for example, it is possible to:

determine the dynamic characteristics of the system

determine the effect of changes under boundary conditions of the system operation


30

optimize metallurgical and other systems, and determine the conditions for their

operation

recommend optimisation of dimensions and other technical parameters of the

equipment

Modelling of processes can be divided into two basic directions. The first direction is

represented by the methods of physical modelling which usually address processes running on

the actual equipment and its scale models of real devices and at normal ambient temperatures.

To this end, the theory of physical similarity between two systems is used. Two phenomena

are physically similar if they are described using the same criterion equation, and if the

corresponding criteria of similarity in homologous points are of the same size.

In comparison with mathematical models, physical models define properties of the

modelled system more completely and more reliably. This is based on the fact that the

physical modelling solves tasks in the substance, while mathematical modelling analyses the

structure of the problem. Moreover, during the construction of physical models it is not

necessary to know the mathematical description of the analysed process. On the other hand,

physical models are associated with higher acquisition prices of some models, have limited

applicability of a particular model, and often it is difficult to change the size of the model

parameters, which sometimes makes it necessary to make do with qualitative solution only.

The second method is the mathematical modelling which includes experimental and

statistical models and analytical modelling. Mathematical modelling is based on a

mathematical analogy (similarity) of two different processes. Physical phenomena of different

nature are mathematically similar when they are described by formally identical

(isomorphous) basic equations. The mathematical similarities then imply proportionality

between the corresponding quantities of analogous phenomena.

An example may be transport phenomena, i.e. processes in which momentum

(viscosity), energy (heat conduction) and mass (diffusion) are transferred from one place to

another. All of these processes are related to disordered thermal motion of molecules. If the

analogy condition is fulfilled, i.e. isomorphism of basic equations, then consistent analogy

between the above phenomena ensures that specific formulae for heat transfer by convection

and for molecular diffusion, determined experimentally, are within certain limits identical.

The method of analogy is used to advantage if we can not solve the fundamental

equations analytically.

4.1 Classification of models

There are a number of criteria by which models are divided into groups (Fig. 17). The

basic criteria include classifying models by:

A. The nature of the process on the model:

deterministic - they are characterised by unambiguously assigned causes and their

consequences, i.e. all the variables, constants, and functions in the model are

deterministic (non-random) variables or functions.

stochastic - at least one variable, constant or function in the model is a random

variable or random function, i.e. either the analysed problem, or the method of the

solution are of random character. This procedure is used when we are not able to

derive a deterministic model, or in the application of some special algorithms of

automatic digital control.


31

B. Aspects of similarity (similarity between the original and the model):

physical

physico-mathematical

mathematical

C. Purpose of the model

cognitive

control

Fig.17. Classification of models by different criteria

D. Aspects of external action:

uncontrolled

controlled

E. Processing of the model information

Analog

digital

hybrid

Furthermore, models are classified based on the expression of space and time, e.g. the

following models:

spatially continuous

spatially discontinuous

unsettled, time-continuous


32

unsettled, time-discrete

settled

Or classification of models based on preserving similarity of the model:

complete - complete similarity of the model in space and time

incomplete - partial similarity

approximate - some dependencies are expressed approximately for the model

Physical modelling of foundry processes

33

4.2 Physical modelling of foundry processes

During pouring, solidification and cooling of castings, very complex processes occur

in the mould involving many physical and chemical laws. With today's level of knowledge it

is, however, impossible to create an exact model of these processes. We must therefore use a

model which can be physically and mathematically described, but which is solvable at the

same time. The basic prerequisite is the same physical nature of the model and the product.

Thermo-physical data are of key importance in the model, as they are necessary to calculate

the heat transfer during the foundry process.

The theory of physical modelling recognizes and uses various types of similarities of

systems, either geometric or other similarities, which characterise different physical

phenomena (area of thermodynamics, heat transfer, etc.). Similarity of two systems requires

the similarity of all essential values in the entire volume of both, the model and the product.

4.2.1 Similarity of systems

Shape similarity between two systems is known as geometric similarity. Systems are

geometrically similar if the ratio of the corresponding linear systems in the model and the

product is the same; this ratio is referred to as a constant of similarity (Fig.18). Geometric

similarity is one of the fundamental parameters, which must strictly be adhered to. Where full

adherence to full geometric similarity is not possible, it is at least necessary to comply with

the geometric similarity of the model and the product in the critical and important dimensions.

Fig.18. Geometric similarity

Foundry casting processes, solidification, and cooling of castings is governed by the

laws of hydrodynamics and heat transfer. For this reason, kinematic, dynamic and thermal

similarity plays an important role in physical modelling.

Kinematic similarity expresses the similarity of velocity fields and acceleration

fields. Basically, it is a balance observed between two geometrically similar systems in which

the ratio of velocity in matching locations of the model and the product is constant and in both

systems the direction of velocity or acceleration is the same (Fig. 19).


34

Fig.19. Kinematic similarity

Similarity of forces between two geometrically similar systems in which the ratio of

forces in the corresponding places and times is constant and their direction of action identical,

is referred to as dynamic similarity (Fig. 20). In dynamic similarity, geometric and kinematic

similarity is assumed.

Fig.20. Dynamic similarity

Similarity of temperatures, temperature gradients and heat flows in the corresponding

process times and corresponding locations of geometrically similar systems is characterised

by thermal similarity (Fig. 21.). Thermal similarity must be ensured for modelling of non-

isothermal processes.

Fig.21. Thermal similarity


35

In the stage of casting and solidification, mainly heat and mass transfer occurs. During

cooling, mass transfer is limited due to gradual solidification of the casting and if disregarded

in the physical model, it has no fundamental consequences for the simulation results as it is in

the casting stage. During crystallisation, heat transfer in the crystal - melt system occurs via

conduction, while in the melt, additionally, this happens by natural and forced convection.

Mass transfer in the melt takes place by diffusion and convection of the melt. In the

crystallisation of pure metal, the process is influenced only by the transport of heat, which is

in the solid phase performed by conduction. In the case of a non-pure metal, crystallisation is

also influenced by the atoms of the additives. In a molten metal, the effect of the melt flow

due to the temperature difference should also be taken into account. In the moulding mixture,

heat transfer is affected by conduction in the contact of two adjacent grains. In the spaces

between the grains it is performed by radiation.

The process of addressing these physical processes is based on a complex solution of

equations of fluid mechanics and thermodynamics. The calculation is based on the condition

that the number of equations must be equal to the number of unknowns. Such a system can

then be called a system of balance equations. The quantities of these equations are usually

mass, momentum, and energy.

4.2.2 Physical model equation

Mass transfer

The flow of molten metal until the onset of solidification follows the basic principles

of fluid mechanics. Mass transfer is defined by the law of conservation of mass which is

widely known as the continuity equation. Continuity equation determines the relationship

between the mean velocity of steady flow of incompressible fluid and variable flow cross-

section S.

In a medium, mass transfer takes place mainly by different types of diffusion

(pressure, concentration, thermal and forced). Another way of mass transfer is the transfer via

natural or forced convection, or mass transfer can also occur by turbulent vortices. In general,

mass transfer takes place during non-stationary accumulation or loss of mass, and also in the

transformation of material components of the given medium.

Mass balance in the transmission of the i-th substance is given by the equation

ipremikonvidifi dmdmdmdm ,,,

where

V

ii dVddm

represents the change in mass of the fluid in the elementary volume

This equation can be interpreted the way that the change in mass of the ith substance in

the volume V is equal to the sum of the mass influx of the ith substance by diffusion and

convection, and the influx or loss caused by different (e.g. chemical) transformations over

time dτ.

After adjustments, i-th substance transfer equation of the medium can be expressed as

follows:


36

0,,,

ipremikonvidif

i qqq divdivdiv

Individual members of this equation represent the change and transfers of the i-th

reacting substance of that medium, considering its conversions in unit volume per unit time.

The basic variable in this equation is the partial density ρi.

Since the overall density of the medium in individual points of the considered volume

is equal to the sum of the partial densities of their individual components ρ = Σ ρi, the total

diffuse transfer of the resulting partial densities must be zero. For the condition of

conservation of the total mass per unit volume, it is necessary to fulfill the equation:

0)(

vdiv

Where ρ is the total density of the medium. This equation is called the equation of

continuity of flowing medium, and it is valid on condition that no discontinuities occur in the

flowing mass of this medium. The equation expresses the fact that a change in the mass of a

certain volume over time is defined as the difference between the amount of mass flowing in

and flowing from the volume (- velocity vector).

It is often necessary to modify this equation as

)( zyx vz

vy

vx

The continuity equation can then be expanded as follows:

)( zyxzyx vz

vy

vx

vz

vy

vx

Left-hand side of the equation represents substantial derivatives of density, i.e.

derivatives by time for a distance following the movement of the fluid, in accordance with the

equation defining substantial derivative:

zyx vz

vy

vxDt

D

Then the continuity equation can be defined in a reduced form:

)( vdivDt

D

Continuity equation in this form describes the rate of change of density as seen by an

observer "carried" by the flowing fluid.

A very important special form of the continuity equation, is the form of the equation

for an incompressible fluid with constant density:

0divv

In real solutions, no fluid is absolutely incompressible, but in practice significant

simplification is very often achieved assuming constant density and practically no error.


37

Diffusion equation (Fick's second law of diffusion)

In the case of an incompressible fluid, where the density is constant, the continuity

equation is simplified, taking the form:

0

zyx v

zv

yv

x

This equation is valid for incompressible fluids, even in unsteady flow. A dependency

can also be defined for a time-dependent change of substance concentration, where D is the

diffusion mass transfer coefficient. In general, the diffusion coefficient is a function of

temperature, pressure and composition of the mixture, mainly the size and mobility of the

particles. To estimate the diffusion coefficient D in various specific applications, there are a

number of empirical and semi-empirical relationships. For example, the Stokes-Einstein

equation is used for diffusion in dilute solutions of colloidal particles or polymers

2 1

,6

kTD

r

where k is the Boltzmann constant, T is the thermodynamic temperature, 2 the

viscosity of the dispersant, r1 is the radius of the dispersed particle.

Then the rate of change of the concentration of substances in a given location is

defined as:

iii Dv

2

If the rate is zero, we get the equation

ii D

2

This equation is known as the second Fick's law which expresses a change in the

concentration gradient over time. The second Fick's law allows one to determine the

distribution of the concentration depending on time and distance x from the reference plane

and determines the rate of change of the mass concentration.

This equation is generally used to determine the diffusion of stationary substances or

solid fluids. It is very similar to the heat conduction equation. This similarity is used in

analogous solutions of problems of heat conduction and diffusion in solids.

Equation of viscous fluids motion

During a flow, the motion (Euler) equation expresses the d'Alembert principle of the

balance of forces of mass, pressure and inertia:

spm dFdFdF

When perfect fluids are flowing, there is no internal friction or heat transfer. However,

a vast majority of processes in the actual motion of fluids cannot be described in this way.


38

The interaction of the fluid particles moving at different speeds leads to dissipation of

mechanical energy. As seen from the existence of internal friction and heat transfer, the

processes are irreversible.

If we consider the sum of all forces acting on an elementary volume dV and the

change of momentum, we can build the equation of motion for viscous liquids. Now, in

addition to the external, inertia and pressure forces which are associated with the motion of

the fluid particles, also frictional forces caused by relative movement of the particles will be

taken into account. The equilibrium is given by the vector sum

frpgs FFFF

If we examine flow in the gravitational field under the action of compressive forces,

taking into account the internal friction forces, we arrive at an equation expressing the

conservation of the sum of forces or momentums. In a vector form, the following force

equation can be written

𝜚𝐷 ��

𝐷𝜏= 𝜚 𝑔 − ∇𝑝 + 𝐹𝑓𝑟

In the coordinate axes x, y, z, it is possible to write the Navier-Stokes equation of viscous

medium as follows

This system of equations can be expressed by saying that the change in momentum

(pressure, gravity) is consumed to change the rate of flow in a given volume and cover

friction losses. At constant variables ρ and η, the equation is adjusted as follows

For an ideal liquid (η = 0), it is further simplified into the Euler equation which is used

to describe the movement of the media, in a flowing fluid viscous effects are of great

importance:

By integrating the Euler equations of motion, we can derive the law of conservation of

energy, the Bernoulli's equation which has a very wide use in practice and can be written as.

𝑔 ∙ ℎ +𝑝

𝜌+

𝑤2

2= 𝑐𝑜𝑛𝑡𝑠.


39

This equation expresses the fact that during a steady motion of a non-viscous

incompressible fluid, the sum of potential, pressure and kinetic energy at any point in the

gravitational field is constant. In turbulent flow, pressure, speed and other variables vary in an

irregular way. The motion is of stochastic, random character.

Turbulent flow is so complex that it can not be accurately described mathematically,

even for the simplest of liquids. Because of the irregularity and complexity of turbulent

motion, mean time-smoothed values of instantaneous velocities and pressures are introduced.

The instant speed of turbulent flow wi is therefore broken down into the mean rate and

fluctuation rate wi' according to the equation.

Where

Similar expressions are introduced for pressure, temperature, and other variables used

Continuity equations and the equations of motion, given for laminar flow of actual

fluid are also applicable to turbulent flow. These equations can not be solved for turbulent

flow, therefore it is necessary to modify the equations so that they describe the time-smoothed

distribution of velocities and pressure.

The time-smoothed equation of continuity for an incompressible fluid in a component

form is:

And time-smoothed equation of motion in the direction of axis x.

The equations of fluid motion in the direction of the axes "y" and "z" are analogous.

The time-smoothed velocities and pressures have replaced instant components. In

addition, new members appeared in the equation of motion that are associated with

fluctuations of turbulent velocity. Expressions of the type can be considered as

additional stresses caused by the turbulence added to the viscous stress and are called

Reynolds stresses. If we wanted to get a description of velocities, we need to substitute them

with an expression.

To express them, various semi-empirical relationships are in use (turbulent viscosity,

Prantl's mixing length, etc ...) Determining the parameters used and their influence in various

regions of flow belongs to the main tasks of experimental research into turbulent flow.


40

Energy transfer

Energy transfer can occur in different conditions. In solid bodies, energy transfer takes

place by heat conduction. In mobile environments, in addition to heat conduction, energy is

also transferred by flow of mass of the medium in space. This type of transfer is called

convection. Apart from conduction and convection, energy can also be transferred by

radiation and other forms of energy. All the mentioned types often occur simultaneously.

When a mould is interacting with molten metal, the mould removes heat from the

metal - the temperature of the mould rises and the metal temperature decreases. When the

temperature of the metal drops to the temperature of metal solidification, it leads to a

transition of the metal from the liquid state to the solid state. This process happens gradually

from the walls of the mould or core toward the thermal axis of the casting. The faster the

mould removes heat from the metal, the faster the crystallisation process is. These aspects

have an impact on the overall character of crystallisation, and hence the properties of the

casting. Because the rate of heat transfer from the metal through the mould is directly

dependent on the thermo-physical properties of the mould, it can be concluded that the rate of

solidification is dependent on the physical and geometrical properties of the casting and the

mould.

The conditions of heat transfer from the metal into the mould are constantly changing

during the pouring, solidification, and cooling process. During the filling of the mould,

transfer from the metal to the mould occurs immediately by interaction of the flowing metal

and the mould walls. When the pouring is complete, the liquid metal is still for some time in

direct contact with the mould walls. After creating a layer of solidified metal at the mould

walls, conditions for the heat transfer from the casting into the mould wall change, as the

mould wall is in contact with the layer of solidified metal. Heat removal from the liquid metal

into the mould therefore proceeds over the solidified metal layer whose thickness increases

with time.

After forming the layer of solidified metal, a gap between the casting and the mould

walls is created as a result of shrinkage. From this point, heat transfer from the melt into the

mould takes place through the already solidified metal layer, but also across the gap. The

thermal conductivity of the gap is lower than the thermal conductivity of the mould and the

solidified metal. This has the effect of reducing the intensity of heat transfer from the melt.

The gap grows depending on the shrinkage of the casting and its thickness depends on the

shrinkage of the metal and dimensions of the casting. Dissipation of heat from the metal

through the mould is a non-stationary thermal process. The temperature in each point of the

casting-mould system is variable with time. To solve these transient processes, it is necessary

to find dependence of the temperature and amount of transferred heat by time for any part of

the body.

The following thermal processes occur in the casting-mould during the filling,

solidification and cooling of the casting:

Heat conduction in liquid metal

Heat conduction in solidified metal

Heat transfer from the melt into the mould

Heat transfer from the solidified metal into the mould

Heat transfer from the molten metal to the solid metal


41

Heat transfer from the solidified metal into the mould across the gap

Heat transfer through the mould

Heat radiation via the inlet system and open risers

In a general case, the energy balance can be expressed by the equation

𝑑𝑄𝜏 = 𝑑(𝑄𝑑𝑖𝑓 + 𝑄𝑐𝑜𝑛𝑣 + 𝑄𝑟𝑎𝑑) + 𝑑𝑄𝑠𝑜𝑢𝑟𝑐𝑒

The change in the total internal energy in the volume V over time dτ is equal to the input of

enthalpy by diffusion, convection, radiation and the total energy from the sources.

The change in the total internal energy is made up of the change in the internal energy

of medium in the volume V, the change in its kinetic energy, potential energy changes of

possible medium transformations and changes in the radiant energy Urad in the volume V:

V

rad

V

ii

VV

V dVdUdVEddVw

ddVTcddQ )()2

()(2

Energies of sources can be expressed by the following integral:

𝑑𝑄𝑠𝑜𝑢𝑟𝑐𝑒 = ∫ 𝑞𝑠𝑜𝑢𝑟𝑐𝑒

(𝑉)𝑑𝑉𝑑𝜏

𝑉

where qsource (V) is the power density of all internal energy sources in a given volume V.

The relationship can be adjusted to the form of a partial differential equation which

describes energy transfer in homogeneous fluids and solids. In most applications, however, it

is not solved in its complex form, but is simplified based on the process being solved. These

modified forms of energy transfer equations are the starting equations for solving various

practical problems of heat transfer.

For example, the equation of non-stationary combined heat transfer by convection and

conduction with internal heat sources is:

𝜕(𝜚𝑐𝑇)

𝜕𝜏−

𝜕𝑝

𝜕𝜏+ 𝑑𝑖𝑣 (

𝑤→𝜚𝑐𝑝𝑇 − 𝜆∇𝑇) = 𝑞𝑠𝑜𝑢𝑟𝑐𝑒

Where the medium is in rest (w = 0), we obtain the equation of heat transfer by

conduction

𝜕(𝜚𝑐𝑇)

𝜕𝜏+ 𝑑𝑖𝑣(−𝜆∇𝑇) = 𝑞𝑠𝑜𝑢𝑟𝑐𝑒

And for each coordinate "x", "y", "z", it will be as follows

𝜕(𝜚𝑐𝑇)

𝜕𝜏=

𝜕

𝜕𝑥(𝜆

𝜕𝑇

𝜕𝑥) +

𝜕

𝜕𝑦(𝜆

𝜕𝑇

𝜕𝑦) +

𝜕

𝜕𝑧(𝜆

𝜕𝑇

𝜕𝑧) = 𝑞𝑠𝑜𝑢𝑟𝑐𝑒

This form of the differential equation of heat conduction is a mathematical description of a

change in temperature over time at any location in the body caused by a transfer of heat and

an action of energy. If the thermal conductivity is constant and no sources of heat are defined,

the equation will be:


42

Or

Stress and strain

In general, it can be said that the action of external forces or temperature results in the

formation of inner forces in the body. The intensity of the internal forces is called stress

whose components can be organized into a stress tensor. The relationships expressing the

component and moment equilibrium equations, which are subjected on the surface S to force

effects, and in the volume of the body to volume forces, are as follows:

In addition, to solve problems of elasticity it is necessary to determine the change in

the body shape using components of deformation, which in general case can include

geometric change the body shape. The link between stress and deformation description

specifies the behaviour of the body under the action of external forces

4.2.3 Dimensionless parameters

Expressing similarity of two systems using the constants of similarity is in practical

terms not very widespread. A more common method is the use of dimensionless parameters in

order to express the similarity between two systems.

A dimensionless parameter has the same value in homologous points in similar

systems, meaning it does not change (1st Similarity Theorem), but has not a constant value

in all points of the systems. In the field of application of similarity theory and modelling,

dimensionless parameters are called similarity criteria and they generally have a specific

nomenclature. Most of these criteria can be expressed using a suitably chosen ratio of selected

forces acting in the system.

Most physical systems can be described using the complete physical equation which is

characterised by taking into account all relevant variables, i.e. values that have significance in

the system. The unification of the complete physical equation with the conditions of clarity

results in obtaining the basic equation whose solution to describe physical phenomena is

usually time consuming, hence difficult to solve. For this reason, criterion equations are used

where the relevant variables are replaced with the criteria of similarity which are derived from

these relevant parameters (2nd Similarity Theorem). Functional dependences between

dimensionless parameters are determined experimentally by measuring the model.

The general form of a criterion equation can be derived using a dimensional analysis

or an analysis of differential equations describing the process.


43

4.2.4 Establishment of criteria of similarity using a dimensional analysis

Dimensional analysis is used when mathematical description of the process is not

known and there is only an assumption that the studied process is a function of the relevant

physical quantities. The dimensional analysis is based on the Buckingham theorem ( -

theorem).

It is based on the principle that any dimensionally homogeneous equation can be

transformed into mutually independent dimensionless parameters generated by an appropriate

grouping of the variables. Mutual independence means that any dimensionless parameter can

not be expressed by the product of parameters raised to different powers.

The principle of this method is shown in the following simple example. A defined

body immersed in a flowing fluid is subject to the force F which depends on the speed of the

fluid's flow w, its density , dynamic viscosity and characteristic dimension l. It is evident

that to express these five relevant variables, we will need only three basic variables, i.e. the

length [m], time t [s] and mass m [kg]. The difference between the relevant and basic

variables suggests that to describe the process, we will need two dimensionless criteria, that

we will mark K1 and K2 and express in a general form:

321

1

aaa lwFK

321

1

bbb lwK

Both equations will be expressed using the basic variables in the dimensional form:

323112

1 )()( aaa mmkgsmsmkgK

3231111

2 )()( bbb mmkgsmsmkgK

For both parameters to be dimensionless, there must be a condition that the sum of the

dimensional exponents for each basic variable must be equal to zero. Therefore, let us create a

system of equations using the dimensional exponents:

For m: 0 = 1+a13a2+a3

For s: 0 = -2 –a1

For kg: 0 = 1+2a

The solution to these equations will give us:

a1 = -2; a2 = -1; a3 = -2

Dimensionless criterion K1 will then take the form:

222

212

1w

p

lw

FlwFK

This dimensionless criterion is known as Euler's criterion Eu.

Analogy can also be used for the second criterion where we obtain:

lwlwlwK

111

2

Which is the reciprocal of the Reynolds criterion - 1/Re


44

Following these steps, the original function, which consists of five basic variables, can

be transformed into a combination of two dimensionless criteria K1, K2. as follows:

0),(),(221

lww

pfKKf

With regard to the studied force F then:

)(22 lw

flw

F

or , in other words, this equation expresses the fact that the Euler's criterion is a

function of the Reynolds criterion.

This example highlights the advantages and disadvantages of the method for the

determination of dimensionless parameters using the dimensional analysis. Linking relevant

variables to form known and proven dimensionless criteria is very useful, however, this

procedure does not find the relationship between the variables.

4.2.5 Establishment of similarity criteria using the similarity transformation method

If the given problem can be described by a basic equation, the dimensionless criteria

can be derived using methods based on these equations. Basically, there are two methods to

analyse these equations and determine the dimensionless criteria.

Similarity of transformation method

The essence of this method is shown by analysing a differential equation of the flow of

a real viscous fluid:

)()(2

2

2

2

2

2

z

w

y

w

x

w

x

pgw

z

ww

y

w

x

ww xxxxz

xy

xxx

For any system similar to the basic system, this equation can be expressed by the

similarity constants, performing a similarity transformation equation which is as follows:

)(

)(

2

2

2

2

2

2

2

2

z

w

y

w

x

w

M

MM

x

p

M

MgMM

wz

ww

y

w

x

w

M

MMw

M

MM

xxx

l

w

l

p

xg

zx

yxxwxw

Where q

qM q

´

q´- value of variable q in the model

q - value of variable q in the product

Both of the above equations are identical, i.e. to maintain similarity of processes when

the resulting complexes of similarity constants for each equation member are identical, i.e.:

2

2

l

w

l

p

g

ww

M

MM

M

MMM

M

MM

M

MM


45

After modifying it this equation takes the form:

122

wlw

p

w

lg

w

l

MMM

M

MM

M

M

MM

MM

M

These dimensionless complexes consisting of similarity constants of each variable are

referred to as indicators of similarity. For similar processes, these indicators are equal to one.

(first theorem of similarity).

The dimensionless parameters in the above example of the flow of a viscous liquid can

be obtained by adjusting the indicators of the similarity equation:

1´´

´

w

w

l

l

MM

M

w

l

then Ho

l

w

l

w

´

´´

This dimensionless parameter is known as the criterion of honochronism. The other

parameters can be derived in a similar way:

lg

wFr

2

2w

pEu

wlwlRe

i.e. Froude (Fr), Euler (Eu) and Reynolds (Re) criterion.

The entire equation of viscous fluid flow can be expressed using the following

dimensionless criteria:

0Re);;;( EuFrHo

4.2.6 Establishment of criteria of similarity using the method of dimensional analysis

of equations

Previously solved equation of viscous fluid flow can also be used to determine the

dimensionless parameters in a different way. It is evident that all members of this

dimensionally homogeneous equation have the same size, in this case kg.m-2

.s-2

. If this

equation is divided by one of the summands, the equation passes into a dimensionless form

from which dimensionless parameters can be easily determined. The whole equation can be

expressed by physical values:

2

2

)(l

w

l

pg

ww

Dividing by the second member, this equation is converted to a dimensionless form,

thus obtaining an equation whose members represent the various criteria which have been

mentioned above:

lww

p

w

lg

w

l

22)1(


46

4.2.7 Overview of the most widely used dimensionless criteria

Reynolds criterion

The Reynolds number is used to assess the nature of a fluid's flow. This criterion,

which expresses the ratio of inertial and viscous forces, is of fundamental importance in

computation of fluid dynamics (friction in pipes and fittings, mixing etc.). As seen from the

definition, the criterion Re can be determined by the equation:

lwlw

Re

The value Re divides fluid flow to laminar and turbulent flow, with low values of Re

identifying the laminar fluid flow. The critical value Re of the criterion (Rek) at which laminar

flow changes to turbulent flow depends on the shape of the environment in which the flow

takes place and also on the characteristic dimension l.

Froude criterion

This criterion is the ratio of inertial and gravitational forces. It provides an

approximate dynamic similarity of flows dominated by inertial and gravitational forces. The

Froude criterion is defined by:

lg

wFr

2

Euler's criterion

Euler's criterion expresses the ratio of the characteristic value of compressive force

and inertia force (momentum flux), i.e. the ratio of momentum fluxes to pressure forces and

macroscopic flow. It can be defined by:

2w

pEu

The value of this criterion is often sought as it contains the desired variable of pressure

losses, and is basically expressed as dependence on other criteria, such as:

)(Re;

)(Re;

(Re)

MaEu

FrEu

Eu

Strouhal criterion (homochronism criterion)

This criterion is taken as an indicator of time-constant speed of movement of the

system's element. The homochronism criterion can be used to express dimensionless (relative)

time of a motion of the element or to express dimensionless (relative) distance. Criterion Ho

is defined by the relation:

l

wHo

Stokes criterion

The Stokes criterion can be defined as the product of the Eu and Re criteria. In the

case of very slow laminar flow, inertia forces, both in the criterion Re and in Eu are negligible


47

compared to the viscous forces and those induced by pressure difference, therefore the criteria

Re and Eu have no sense. In such a case it is advisable to eliminate the inertia forces. Stokes

criterion is defined as:

w

lp

l

dwStk

2

Weber's criterion

Weber's criterion is defined by the ratio of inertial forces and capillary forces which

are caused by surface tension. For the modelling of metallurgical systems in practice, in some

cases it is necessary to ensure simultaneous fulfillment of this criterion and Fr criterion.

Weber's criterion is defined as:

lwWe

2

Prandtl criterion

Prandtl's criterion includes the properties of fluids which are important in the

molecular transfer of momentum and heat. It can be calculated as a proportion of the Péclet

criterion, which defines thermal conduction in the boundary layer, and the Reynolds criterion.

The Prandtl criterion is defined by:

pc

alva

lv

Pe

RePr

Nusselt criterion

Nusselt criterion defines heat transfer by convection. Basically, it is a dimensionless

heat transfer coefficient and its dependence on the conditions of heat transfer is expressed as a

function of other criteria:

nGrCNu Pr)(

Where C n are constants; Gr - Grashof's criterion defining the natural convection of

viscous fluids, Pr - Prandtl criterion.

The intensity of heat transfer is expressed by the heat transfer coefficient whose size

depends on the properties of the medium, the speed and flow pattern and geometries of the

surfaces involved. The value can be determined by the Nusselt criteria:

CLNu

where LC - is the characteristic dimension (defined by the geometry of the body flown

around)

Mathematic modelling of Foundry Processes

48

4.3 Mathematical modelling of Foundry Processes

A mathematical model consists of a system of abstract mathematical relationships

describing the essential characteristics of the examined object and thus providing a clear

description of all the relevant factors of the situation, enabling one to detect fundamental

relationships between the components of the system under study.

For a mathematical description of the characteristics and behaviour of an object, it is

necessary to determine the variables which describe how the surroundings affect the systems

(inputs) and variables which the system exhibits for its surroundings (outputs). A

mathematical model expresses the dependence of outputs on inputs described by

mathematical relationships. These relationships become a mathematical model only when

they are clearly assigned to the particular process or phenomenon. The process of determining

a mathematical description of a system is called system identification. In the identification

phase, the format of the model should allow its use in the area in which we want to use it.

The process (system) we want to describe mathematically, is governed by certain

physical, physico-chemical and chemical laws which have their mathematical expression.

This interpretation of laws is used in compiling deterministic models. Sometimes we can

describe all the specified conditions and relations of the modelled process in this way,

obtaining a precise mathematical model. However, such a mathematical model is so

complicated that its solution is unfeasible. In addition, usually a full description can not be

obtained because we do not know the process in such details. In practice, however, absolutely

accurate results are not necessary and it is sufficient if the model captures the essential

characteristics and behaviour of the process. We can therefore afford to leave out or simplify

some of the less important influences and relationships. The ultimate criterion for the quality

and usability of the model is always its adherence to the reality in accordance with the

purpose for which it was created. Figure 22 is a schematic of the stages of creating a

deterministic model.

Fig.22. The process of creating a mathematical model


49

When compiling a mathematical model it is necessary to perform:

thorough analysis of the system including the decision about the importance of

the following elements and therefore their inclusion in the model:

– specify processes which take place within the process and determine

their nature

– define the influences affecting the process and its progress

– determine quantities describing the process

This step is used to obtain a theoretical model, which does not describe accurately the

reality, but its advantage lies in the simplicity, clarity and consequently easier handling of the

resulting equations.

Based on the theoretical model, a mathematical description of the process follows.

This step involves the selection of mathematical description of the patterns used in the

theoretical model, creating the model equations, i.e. completing selected links by simplifying

assumptions and the necessary mathematical adjustments, and finally the conditions of the

solution (usually initial and boundary conditions for the solution of differential equations). At

this stage, usually mathematical equations are used expressing the known laws and

relationships of physics, physical chemistry, etc. The result of this procedure is a general

mathematical model of the process.

The third phase of the whole process is the creation of the model. This means creating

a simulation program, where the method of solving the model equations is selected, followed

by their processing in order to find an appropriate solution algorithm. The fourth phase is the

creation of the simulation model. The result is a computer program suitable for use in

practice. This phase includes the following steps:

identification of the model, i.e. finding the values of unknown parameters of the model

(e.g., by comparing the results of the solution obtained with data from the literature, with

the experimental values, etc.)

model verification, i.e. solving control tasks and analysing their results to verify the

accuracy of the model across the entire application area, assess accuracy and model

fitness for the purpose.

The basis of the mathematical description of processes in metallurgy of liquid metals

is a selection of a suitable hydrodynamic model of the process. This is followed by a

description of the physical, physico-chemical, thermal and other processes as a system of

differential equations, including also empirical equations. These equations are usually

calculated by numerical integrations. Mathematical model of a complex system can contain

up to 105 variables and a corresponding number of equations. For very complex systems, a

corresponding model can not be constructed, or the model can not be mathematically solved.

To solve the mathematical model, two methods of solution can be used:

Analytical (explicit) solution lies in finding the exact solution using analytical

mathematical methods (solving systems of equations, solving tasks focused on

extreme case, etc.).

numerical (approximate) solution is used to solve models where the problem

can not be solved analytically or in cases where the analytical solution is

difficult and complex. In the case of numerical solution it is necessary to


50

consider its numerical stability, convergence and error that arises from the

solution

Analytical models are constructed based on a description of the internal structure of

the system, i.e. knowledge of natural laws of processes and equipment design in which the

processes take place. The advantage of these models is the possibility of application to a

wider range of applications. The disadvantage is a complex way of making the model,

computational program and the work is excessively time consuming. Analytical models are

mostly used for smaller and simpler systems.

Mathematic modelling of foundry processes - analytical methods

51

4.3.1 Analytical methods

Analytical methods allow one to obtain a solution to the task in the form of a

mathematical expression for the desired variable as a function of spatial coordinates and time.

The solution must correspond to a certain equation and unambiguity conditions. Most

technical tasks require a simplified mathematical model in order to make the task solvable.

Correct determination of the degree of simplification of the mathematical model while

maintaining its credibility is a major problem in the use of analytical methods.

Typical analytical methods include the method of separation of variables, called the

Fourier method. Another group of analytical methods, integral transformation methods, is

based on the principle of mathematical transformations of variables. The Laplace and Fourier

transformations belong to the most common ones. The use of the integral transformation

methods as well as the classical analytical methods is restricted to linear problems with

boundary conditions and easier areas.

Fig.23. General procedure in mathematical modelling

Other methods used are variational methods. Unlike the previous methods, these are

suitable for approximate solutions of nonlinear problems. Their principle is that instead of

solving a differential mathematical model of the physical field, the solution is focussed on a

variational problem of an extreme of a functional in an integral form characterising the

process.

Usually a minimum of energy functional is involved. Out of several variational

methods, Ritz method is the most famous one. Other analytical methods transform tasks with

boundary conditions to other types of equations and problems, e.g. by using the Bessel

functions, etc.


52

Fig.24. Procedure for modelling a specific case

To solve technical problems, analytical methods are of limited use. Exact methods are

used primarily for control solutions, usually of one-dimensional tasks with simple boundary

conditions, approximate methods are used for more complex boundary conditions as well.

Approximate analytical methods use Laplace integral transformation and variational methods.

The solution is delivered in the form of relatively simple dependency, for example several

members of a series. The accuracy of results is usually sufficient.

4.3.2 Initial and boundary conditions

For models described by differential equations, the description must be complemented

by an appropriate number of boundary and initial conditions (conditions of uniqueness). For

each independent variable, we need as many mutually independent conditions as is the highest

order of derivative occurring in the equations with respect to this variable.

Formulation of initial and boundary conditions is an integral part of creating a

mathematical model. Some conditions arise quite simply from the task assignment (e.g. at the

beginning the temperature at all points is the same and equal to a certain value), others must

be derived in the same procedures as the mathematical model (e.g. based on the balance). As a

check of the correctness of their derivation, we can use the fact that, in general, for a

mathematical form there are only a few types of conditions and therefore, in a particular case,

we have to achieve equality with one of them.

For a description of the mathematical form of the types of boundary and initial

conditions, we will use the following denominations of variables:

u – dependent variable; t – temperature; tp – temperature on the surface of the body

– time,

x, y, z – coordinates,

f - function prescription whose form is known and the value of the function can be calculated

at any time.


53

For clarity, examples will be shown here of the conditions of uniqueness for heat

conduction

The initial condition is usually one, defining the situation at the beginning of the

solution. Generally, it can be expressed as

for = 0: u = f (x,y,z)

For heat conduction: for = o = 0 t = f(x, y, z)

in other words, at the beginning of the process, i.e. at time t0 the dependent variable u

is a known function of coordinates x, y, z.

Boundary conditions occur in cases where coordinates act as independent variables.

There are three basic types of boundary conditions:

a) boundary condition of the 1st type (Dirichlet):

for x = x0 : u = f (x,y,z,)

For heat conduction: tp = f(x, y, z, )

i.e. the value of the dependent variable at the point x0 is a known function of the other

coordinates and time.

b) boundary condition of the 2nd type (Neumann):

for x = x0 :

),,(

0

zyfx

u

xx

For heat conduction:

defines the distribution of heat flow density q on the surface of the body as a function

of coordinates and time

q = f(x,y,z,), thus q = - gradt = - pn

t

first Fourier law

where n is the normal to the surface of the body

or the value of the derivative of the dependent variable by one coordinate (e.g. by x in

point x0) is a known function of the other coordinates and time. We often encounter the

second type of boundary condition in the form

for x = x0 : 0

n

u

i.e. the derivative of the dependent variable with respect to the normal to a surface is

zero.

c) boundary condition of the 3rd type (Newton):

for x = x0 :

,,),,,(

0

0 zyfx

ubzyxua

xx

For heat conduction, it is used in cases where the specified ambient temperature tok

and the coefficient of heat transfer to the surroundings c: then:


54

p

okpcn

tttq

i.e. the value of a linear combination of the value of the dependent variable u in point

x0 and its derivative by x in the location x0 is a known function of the other coordinates and

time; constants a and b are coefficients of the linear combination.

Methods of casting - gravity casting

55

4.3.3 Numerical methods

In technical calculations, it is necessary to know not only the initial and boundary

conditions, but also to know material characteristics of all materials of the system being

solved. In many articles on mathematical modelling in the foundry industry we can find a

sentence that knowledge of thermo-physical data is the alpha and omega of the accuracy of

the results obtained. It is therefore quite clear that the expected result will be as precise as the

input data we use. The basic input data include material viscosity, thermal conductivity,

enthalpy, density and the proportion of the solid phase.

In the calculations of stress and deformations we should add the knowledge of

elasticity moduli, thermal expansions, and more. It should be noted that these data are useful

for the calculation only if they are a function of temperature.

Data can be obtained:

from material database of the simulation program

from the literature, where, however, data is specified mainly for pure elements,

or for basic types of material, and usually for room temperature

direct experimental measurements (expensive)

inverse modelling (combination of experiment and numerical calculations).

Their essence lies in the discretisation of variables, therefore they have a very

significant potential in the application in computer modelling. They are characterised by

repeatability of simple algebraic operations of certain type, which corresponds to the

operating characteristics of digital computers. Numerical methods allow one to get the

solution of problems in a finite number of discrete points (nodes) of the selected difference

network or finite element network in the whole area, or in its surface portion.

Numerical methods are divided into:

finite difference methods (FDM)

finite volume methods (FVM)

finite element methods (FEM)

boundary element methods (BEM)

In the simulation programs of foundry processes, computational modules mostly occur

using the method of finite differences and method of finite elements. For this reason, we will

only give a short description of the two mentioned methods.

Finite difference method (FDM)

The finite difference method is becoming one of the most frequently used methods of

approximate numerical solution of partial equations. It is simple, universal and can be used

for very diverse types of boundary tasks, including nonlinear ones. A large part of the most

important technical problems leading to partial differential equations are therefore solved by

this method.

The principle of the finite difference method consists in approximating the basic

differential equation with the appropriate boundary conditions corresponding to the difference

equation which has a form of algebraic equations. This means that the partial derivatives in


56

differential equations describing the behaviour of the model are replaced by differences, i.e.

linear combinations of functional values of the lookup function in nearby points:

h

afhafaf

h

)()(lim)´(

0

f´(a) - derivative of the function value in point a, f(a) - functional value in point a, h - step

length

Step h is often replaced by the time change, this form is called the forward difference.

The functional value of the derivative can be expressed as:

h

aR

h

afhafaf

)()()()´(

where R(a) specifies the measurement error not included in the calculation. This

inequality is referred to as a discreditisation error. The total inaccuracy of the calculation is

the sum of this deviation and a rounding error.

The more accurate expressions the derivative uses to substitute, the more perfect the

approximation is. This difference approximation is called "explicit difference scheme". The

substitution is performed at discrete locations formed by nodes covering the studied area. The

end result of algebraic operations is to determine the desired value in the node.

Derivation of an example solution will be performed as a general task in two spatial

dimensions:

where

and for

The initial condition (base)

Boundary conditions (the side walls)

Grid equations are derived, where N, r - are

natural numbers, h = (b-a)/N; r

T

In the region, a grid is specified by nodes (xi, yj,

tk)

tyxfy

u

x

utyxfu

t

u,,,,

2

2

2

2

),,,,,0 baxbayxTt

yxyxgyxu ,,,0,,

hranicinayx

Tttyxtyxu

,

,0,,,,,

Niihaxi ,...,0,

Njjhay j ,...,0,

rkktk ,...,0,


57

For fixed k, a set of points (xi, zj, tk ) will be called k - th time layer

the forward difference is substituted with derivative by time t

Where )(

,

k

jiu is the approximate solution of the problem in the node (xi, zj, tk )

A derivative by variables x and y will then be used to substitute as follows:

Derivative by x using the values

Derivative by y using the values

On the k-th time layer, then:

And the subsequent derivatives will be:

On the whole, therefore, the equation can be modified using differentials as follows:

k

ji

k

ji uu ,

)1

,

k

ji

k

ji

k

ji uuu ,1,,1 ,,

k

ji

k

ji

k

ji uuu 1,,1, ,,

h

uu

x

uk

ji

k

ji ,1,

h

uu

y

uk

ji

k

ji 1,,

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuuh

h

uu

h

uu

x

u

,1,,12

2

,1,

2

,,1

2

2

21

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

uuuh

h

uu

h

uu

y

u

1,,1,2

2

1,,

2

,1,

2

2

21

k

ji

k

ji

k

ji

k

ji

k

ji uuuuuhy

u

x

u,1,1,,1,122

2

2

2

41


58

Fig.25. Visualization of the explicit method

The values on the k-th time layer are used to calculate the new value ui,j on the (k+1)-

st layer. This is called the explicit method through which we receive directly recurrence

relations and there is no need to solve a system of equations. If we want to obtain a

convergent and numerically stable method, the implicit method is used which uses a

feedback differential, and a second-order differential at the same time.

When replacing the derivative by the time t with forward difference, analogous to

derivative by x and y values and a substitution using values on (k+1) time layer:

It will result in an analogous relationship

This yields a system of equations (implicit method). This method is numerically more

challenging. This deficiency can be compensated by using large time steps.

Fig.26. Visualisation of the implicit method

A combination of the implicit and explicit methods (linear combination) is represented

by the Crank-Nicolson method which uses a difference in time n+1/2 and again a central

difference of the second order. This method is always convergent and numerically stable. The

size of the deviation increases depending on the distance of the point from the edge, and

therefore it is necessary to use a time-fine grid.

1

,1

1

,

1

,1 ,,

k

ji

k

ji

k

ji uuu 1

1,

1

,

1

1, ,,

k

ji

k

ji

k

ji uuu

11

,

1

1,

1

1,

1

,1

1

,12

,

1

,4

1

k

i

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

jifuuuuu

h

uu


59

Fig.27. Crank - Nicolson method

The resulting solution of the model equation will be:

The general procedure of the FDM then proceeds in the following steps:

Selecting a suitable set of nodes (selection of the grid) - the finite difference method

is used only for closed areas with known boundary conditions at the border area. Creating a denser grid requires a more accurate calculation, but takes up more memory

in the computer and the calculation is longer. We can also choose a variable density of the

grid. Where the value of the monitored function changes more, a denser grid can be defined

and for the rest of the geometry, the grid will be coarser.

Fig.28. An example of grid with different nodes

Approximation of differential operator by difference operator

Is the already mentioned substitution of a differential equation with a difference

equation. The subsequent solution is calculated only in the nodes of the defined grid.

Development of a system of equations (boundary conditions) Furthermore, these equations are compiled into a calculation matrix and allocated to

each node of the coordinate.

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

k

ji

ffuuuuuh

uuuuuh

uu

,

1

,,1,1,,1,12

1

,

1

1,

1

1,

1

,1

1

,12

,

1

,

141

4


60

Solving the system of equations When solving the equations we use Gaussian elimination, eigenvalues-vectors or

iterative methods.

Finite difference methods can according to the selected type of difference expression

be divided into explicit, implicit, and combined ones, and may be implemented as a single

layer, multi-layer or multiple.

Efforts to reduce the extent of computing operations when solving various complex

and multiple-dimensional tasks leads to the creation of increasingly more economical

difference expressions which are suitable for high-performance digital computers. The basic

characteristic properties in the process of assessing the methods include convergence,

accuracy and stability of the solution.

The grids can be divided into square, rectangular and regular or irregular, and special-

purpose grids, such as hexagonal, polar, etc. Today, the most common types are rectangular

grids. Irregular grids are used to facilitate the formulation of the boundary conditions and to

densify or dilute grids because the accuracy of approximation depends upon the density of the

grid. When densifying a grid, numerical calculations become more demanding, therefore it is

advantageous to compress the grid only in those places where we want an increased accuracy.

The advantage of FDMs is the simplicity of programming and numerical

implementation and the relative simplicity of nonlinear mathematical models. On the other

hand, the disadvantage is in the problem of approximation of boundary conditions at different

parts of the borders which are not applicable to the different density grids, and also

deterioration in the accuracy of the approximated solution for a grid with different spacing of

nodes.

Fig.29. A grid generated by the FDM method

From the above facts, it is evident that the grid method is basically applicable to any

type of partial differential equation. For some types of tasks, however, it is often necessary to

use only a special type of grid where the time division is dependent on the spatial division.

Finite Element Method (FEM)

The finite element method is one of the variational methods. These methods were

developed based on the discovery of the Dirichlet principle for solving differential equations.

These methods are based on the choice of solution to the problem from a group of possible


61

solutions. In variational methods, we look for solutions to the task using experimental

solutions. A continuous region is divided into finite elements of suitable shapes bonded

together at the nodes.

Continuity of the functional and the distribution of values in the elements is

accompanied by discontinuities on their borders. Therefore, to determine the looked for

dependence, it is advisable to use an integral functional. It is represented by an integral over

the entire area and part of the boundary at which the respective functions of temperature are

not known, or through their derivation we can proceed by expressing the functional as a

function of the expected experimental solution. Then, of all the possible solutions satisfying

the boundary conditions, we select the one that makes the functional stable - ensuring its

minimum.

If we have a partial differential equation:

02

2

2

2

y

u

x

u

Which is defined in a limited region G with a known boundary condition on the border

as: u=g(s)

The following Dirichlet integral can be compiled:

G

dxdyy

u

x

uuI

22

By finding a function which minimises this functional, the solution to the differential

equation has been found as well

If a functional is given, for example as follows:

2

1

´´)´,,,(

x

x

dxyyyxF

defined in a closed interval <x1,x2> with prescribed function values at endpoints: y =

y1 and y = y2, then the function y in Fig. 30 is the exact solution to the problem. The

variational method looks for a solution close to it. Two such solutions are marked in the

picture as

1y and

2y . Any of such experimental solutions can be expressed using a function

describing the exact solution and its variations y. Then an equality applies:

yyy

Variation of the function y = y(x) is defined as an arbitrary infinitesimal change of

function for a given value of the independent variable x.


62

Fig.30. Variational method of solving the value of function

The method of finite elements is based on dividing the body into finite elements. A

finite element which is the basis of this method, is a selected element (volume, area, length)

defined by nodes in the corners, or on the edges. In this way, an indiscreet body is converted

to a discrete body consisting of elements which are interconnected in the nodes - in a finite

number of points.

In elementary areas, the solution of differential equations is approximated by simple

functions - linear or quadratic polynomials. The initial partial differential equation is

converted to a system of linear algebraic equations for the desired values of the potential in

the nodes.

The general procedure of the FEM method can be divided into the following steps:

Discretisation of the analysed area

Dividing the analysed area into subareas (finite elements - elements), which have the

following properties:

they do not overlap

their unification covers the whole analysed area

in each grid element constant parameters of the analysed structure

They can take a shape of:

o lines (1D)

o triangles (2D)

o rectangles (2D)

o tetrahedra ( 3D)

The simplest element for planar tasks is a triangular element with three nodes which

approximates the looked for function by a linear polynomial with parameters a1, a2, a3 as

follows:

yaxaau 321

Approximation of the desired function


63

The physical properties of the body, such as displacement, stress, temperature, etc.,

can be replaced with the function of spatial coordinates. This function is called the

approximation function, or shape function.

Development of matrix equations Solving matrix equation (using the inverse matrix, Gaussian elimination)

Using the FEM (finite element method) allows one to solve huge systems of equations,

with up to millions of equations and millions of unknowns on computers with parallel

architecture.

Fig.31. A grid generated by the FDM method

Comparison of FDM and FEM methods

The answer to the question of which of these methods is preferable is not clear.

Generally, FDM allows easy discretisation, which means less hardware of the computer and

shorter calculation time. On the other hand, interleaving the grid by a geometrical model

deforms the rounded or chamfered parts of the model (in particular in cases with larger

spacing of the grid points), resulting in lower accuracy of the calculation, unless local

correction is performed. Some methods of solution have a built-in algorithm which during the

calculation automatically performs the correction of volume and surfaces of elements

depending on the actual geometry and the model materials used.

The advantage of the FEM method is that it better tracks the geometric shape of the

surface of the model, allows increased local density, i.e., selecting a greater grid density in

some hot spots. Unlike FDM, casting deformation in the interaction with the mould can be

solved, as the introduction of non-linearities of a large deformation type and contact

conditions (temperature and deformation) is relatively simple for FEM. The disadvantage is

seen in greater demands on computer hardware and longer calculation time. Nevertheless,

both methods can also be combined. For example, the process of casting i.e. calculations of

flow and heat transfer can be solved using the FDM method, followed by the solution of

strength and deformation problems using FEM. In this case, however, it is necessary to

transfer the values from the FDM grid nodes to the FEM nodes.


64


Physical modelling

Mathematical modelling

Similarity

Geometric, kinematic and dynamic similarity

Criteria of similarity

Analytical Solutions

Numerical solution

Conditions of uniqueness

Finite difference method (FDM)

Finite element method (FEM)


1. What can be solved using modelling for the relevant industrial equipment?

2. How do we divide the modelling of processes?

3. What are the main differences between the physical and mathematical modelling?

4. What is geometric similarity?

5. What is kinematic similarity?

6. What other similarities between the model and the modelled system must be met so that

kinematic similarity is valid?

7. What is dynamic similarity?

8. What other similarities between the model and the modelled system are expected so that

dynamic similarity is valid?

9. Which basic equations are solved in the physical modelling of casting processes?

10. What is a dimensionless parameter (similarity criterion)?

11. What does the first theorem of similarity say?

12. What is the basic equation?

13. What does the second theorem of similarity say?

14. What is of key importance in a model for the calculation of the heat transfer during a

foundry process?

15. What methods of solving a mathematical model can be used, what are the differences?

16. What are conditions of uniqueness?

17. How many types of boundary conditions are there, and what are they?


65

18. What numerical methods are most commonly used for the simulation of casting

elements?

19. What is meant by discretization of the analysed area?


66

5 Numerical simulation

5.1 Architecture of simulation programs

Study time: 2 hours


define the relationships of heat transfer process dynamics

define each step of the simulation calculation

describe the architecture of simulation programs

Presentation

In recent years, numerical simulation of casting processes achieved a strong position

among the tools used to optimize casting production technologies. Although the

technologist’s experience is still irreplaceable, they can use the simulation software as a

powerful tool, allowing them to optimise processes, increase metal utilisation and reduce the

percentage of non-conforming products and thereby to streamline production. The aim of

mathematical modelling is to tune the designed technologies in the preparation phase of

production to avoid expensive experimental testing.

At a glance, all available simulation programs differ in their graphical arrangement of

user interface, but their architecture is very similar to each other, irrespective of operating

systems in which they run (Fig. 32).

Fig.32. Main stages of simulation calculations


67

Crystallisation of castings is controlled using the heat transfer control in the system of

casting - mould - ambient environment. Excessive heat from liquid metal and crystallisation

heat must be removed from the cast metal, based on the following equation:

Δ𝑄𝑚𝑒𝑡𝑎𝑙 = Δ𝑄𝑚𝑜𝑢𝑙𝑑 + Δ𝑄𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑

The dynamics of the heat transfer process depend on:

Geometric arrangement and weights of the mmetal and mmould components

Method of filling the mould with liquid metal

Initial temperatures of all system components and their spatial distribution

Crystallisation intervals TL and TS , and the metal crystallisation heat Qcrystallic

heat conduction coefficient metal;mould

Specific heat capacity cmetal; cmould

Densities metal;mould

Conditions of heat transfer from the metal to the mould which are defined by the heat

transfer coefficient k-f

Cooling conditions on the outer surface of the mould expressed usually as the

surrounding medium temperature TOK, and the coefficient of heat transfer into f-ok

For this reason, the operations of simulation calculations are usually preceded by a

database of metal and moulding materials. Proper simulation results applicable in practice are

based on the knowledge of thermo-physical properties of the moulding and casting materials.

Using improper values of the required data is the most frequent cause of differences between

the results obtained from simulation and experimental measurements under comparable

conditions.

If the database does not include the material actually used in a specific case, it must be

added, e.g. experimentally, and adjusted based on the specific conditions.

Basic thermal properties α, c, λ, ρ, etc. should be defined based on the temperature.

Fig.33. Main stages of simulation calculations


68

5.1.1 Preprocessing

In this step, geometric data of castings is created. The geometric data is entered in the

program in two ways. The data is either transferred from external CAD systems in various

export formats (.stl, .ogs, .dxf, .iges, etc.), or a complete geometry of the casting is created

within the Geometric Functions of Network Generator simulation program which helps the

CAD system prepare a design for the spatial simulation of the casting solidification.

In some simulation programs, the geometry created in CAD application must be

converted in the FEM generator, and only then to load this format in the simulation software.

This mainly includes defining areas, checking the generated network and correcting it.

Sometimes, the mould shape must be additionally defined in relation to the generated casting

grid.

Fig.34. Model of a casting imported into the Pam-QuikCAST simulation program

Further in this step, materials are assigned to each item (metal, mould, core, chills,

riser lining, filters, etc.). The pre-processor is also used to define the mould size and the

casting’s position in the mould, for specify the cast metal materials including the casting

temperature from the database, the mould material and/or cores and their initial temperatures,

etc. The studied area is divided into sub-areas in which computation is then performed. The

grid density has an impact on the calculation accuracy, but also on the computation time and

computer hardware requirements (RAM), and steps and data which should be stored on disc

are specified.

Boundary and initial conditions are defined (temperatures, speeds, pressures, etc.) and

the necessary expansion of the surface / volume of the object takes place here for one of the

discretisation method of the solution (FDM, FEM, etc.).

Setting the boundary and initial conditions is essential for a correct simulation. Always

use a specified technological procedure and maintain all parameters listed in the procedure.

These parameters can include the pouring temperature, mould temperature, pouring height,

properly defined transfer conditions and heat dissipation, defining the mould thickness, mould

permeability, surface roughness (average thickness of grog), etc. This depends on the type of

the simulation software.

Some software provides a very easy way of entering the initial conditions (this may

only include the definition of heat transfer to the outer walls of the mould only) which can be

very complicated in other software (in the geometry preparation phase, x areas can be defined


69

and several initial and boundary conditions can be entered for each area - sometimes even

time-dependent variables). The next important step includes defining the point/area where the

metal enters the gating system, and assigning the initial conditions to the given volume. To

determine the temperatures in different parts of the casting and mould, placement of

"imaginary" thermocouples can be defined.

5.1.2 Mainprocessing

This is the main step of the simulation calculation. It is the computation module itself

which, in predefined increments, saves selected data of numerical solution of the defined

simulation model on the hard disc. The program calculates the temperature changes during the

simulated moulding process, and/or stresses or a micro-structure are analysed.

5.1.3 Postprocessing

It is used for retrieving calculated data sets, browsing, viewing and studying the

calculated and saved data sets. They allow one to monitor and analyse velocity, temperature

and pressure fields during filling the mould cavity, the solidification process, formation of

shrinkage, solidification time. More advanced software allows the observation of internal

stresses (tensile and compressive stresses) and residual deformations, and the prediction of the

casting’s structure and microstructure.


Simulation program architecture

Preprocessing

Mainprocessing

Postprocessing


70

6 The use of simulation programs for different methods of casting

6.1 Gravity casting

Study time: 1 hour


define the fundamental principles of gravity casting

define areas preferably monitored during gravity casting simulation

Presentation

Production of castings using the gravity casting process is one of the basic casting

production methods. The mould is by the effect of the weight of the molten metal. There are

two methods of gravity casting.

The first method is the gravity casting in sand (non-permanent) moulds, allowing the

production of any casting, regardless of its complexity, shape, dimensions, weight and

material. However, this method is associated with a lower dimensional accuracy, lower metal

utilisation, etc. The other method is the gravity die casting.

It is a quite simple technology. Moulds are usually made by casting and are made of

spheroidal graphite cast iron. The parting plane, clamping risers and ejecting holes are

machined, while the functional area of the mould cavity often remains in the cast state. The

advantage of this technology is that solidification is faster than in sand moulds. Castings

produced in this way have finer grained structure, better surface quality, etc. On the other

hand, cost of making moulds are significantly higher and this method is only used for certain

types of alloys.

Regarding the use of simulation programs, the following items are most frequently

considered for the gravity casting:

The character of filling the mould cavity with respect to the alloy type and geometry

of the designed inlet system,

Solidification method, the location of thermal axes and nodes, and the associated

defects such as shrinkage and shrinkage porosity,

Location and efficiency of process technology allowances (dimensions and efficiency

of risers)

Calculation of solidification and/or cooling of castings, what residual stresses can

remain in castings, whether it is prone to cracks and fissures and how the casting will

be deformed


71

Generally, filling the mould with liquid metal is governed by the laws of

hydromechanics; speed of filling the mould cavity, flow type, whether the mould cavity

filling is easy, i.e. laminar, or whether the filling speed is too high and the stream of liquid

metal is of turbulent nature and leads to the subsequent oxidation of the metal.

Fig.35. Visualization of mould filling with liquid metal

When designing new technology procedure, we try to prevent the formation of casting

defects such as shrinkage and shrinkage porosity, or to minimise or relocate them to make

them acceptable. This type defects are present in most technical alloys and their character is

associated with the reduction of volume known as shrinkage, which occurs during the cooling

and solidification of the melt. To prevent occurrence of shrinkage in castings, it is necessary

to replenish the volume deficit from sufficiently dimensioned risers based on repeated

simulations.

Numerical simulation can help answer these questions. The first calculation of

solidification of the casting itself (without risers, inlet system) suggests how the casting

solidifies, where thermal nodes are formed and where the last solidification locations are. This

analysis helps in the design of risers layout and/or other elements affecting the thermal

conditions during solidification and cooling (chills, insulation). This is followed by a re-

preprocessing phase, during which the technologist prepares a draft of sizes and locations of

risers and/or other parts. To obtain a more accurate idea of the heat balance of the system in

question, it is suitable to model the mould filling.

Fig.36. Visualisation of the casting solidification


72

After obtaining the results of filling analyses, the solidification and/or cooling of the

casting are calculated (stress and deformation). The nature of the temperature field during

solidification can be used to observe whether a directed solidification occurs and whether the

risers are sufficient in terms of heat and volume. The progress of solidification front with the

optional use of special criterion functions can be used to determine the feeding distances of

risers. We can directly see whether there is any independent solidification of some parts of the

casting, which are separated from the additional feeding of the liquid metal from the risers.

Fig.37. Views of solidification and porosity in the casting: prediction of shrinkage in the

casting

If the selected technology procedure provides a satisfying solution to the problems

associated with solidification, we can further optimise e.g. the sizes and types of the risers.

Nowadays, it is common to use insulating or exothermic lining. These modern technological

aids can also be included in the calculation.

In complex castings which have complicated transitions of walls, we often wonder

what residual stresses may remain in the casting, how much they are prone to cracks and

fissures and how the casting will deform. Material models include elastic, elastic-plastic or

elastic-viscous properties of the casting or mold. During the calculation, breaking the

solidified surface of the casting from the mould is taken into account and therefore also the

formation of air gaps. Heat transfer coefficient is recalculated automatically, enabling the

accurate calculation of heat transfer during the solidification and cooling process. For rigid

moulds, the impact of retarded shrinkage on the occurrence of stress in the casting and/or its

subsequent deformation after the removal from the mould can be observed. These calculations

specify the causes of adverse events and encourage the ideas of changing the heat balance of

the process and/or they are the reason for the part geometry modification.


The process of filling the mould cavity

Solidification of castings

Shrinkage and shrinkage porosity

Porosity.

Methods of casting - shell casting

73

6.2 Shell casting

Study time: 1 hour


define fundamental principles of shell casting

define the areas which are preferably monitored during precision casting

simulations

define the shell casting conditions and processes which are different from

the other conventional methods of casting

Presentation

Generally, it is necessary to remove excessive heat from the liquid metal and the

crystallisation heat from the cast metal.

In the case of shell moulds, this heat is partly accumulated in the shell mould and

partly transferred into the surrounding environment. In this aspect, the thermal processes

during shell casting are different from other foundry technologies. Unlike the commonly used

moulds made of dispersion materials (sand moulds), during casting in annealed, relatively thin

moulds the amount of heat transferred into the surrounding environment is important.

Temperature mode of the mould consists of the following steps:

annealing the shell

transport from the annealing furnace to the casting field

delay in the casting field before starting the casting

dissipation of heat from the mould after casting the liquid metal

The initial temperature profile arises at the moment of removing the mould from the

annealing furnace, where there is a homogeneous temperature field and cooling occurs. In this

phase, heat is dissipated mainly by radiation and convection to the surrounding environment;

these conditions are relatively difficult to define and forced cooling resulting from the

movement of the mould and air also play a role in this process. During the mould standing in

the casting field until the time of pouring, heat dissipates to the surrounding environment by

convection and radiation.

Therefore, it is necessary to solve the course of metal cooling and solidification

simultaneously in all the above stages. The amount of accumulated heat depends on the

metal/mould weights ratio and on the initial temperature of the mould. For casting in moulds

with a high initial temperature (after annealing), the significance of the heat accumulation


74

further decreases. Thermal accumulating capacity of shell moulds is important for thin-walled

and large-shaped castings with short solidification times. For thick-walled compact-shaped

castings, the portion of heat dissipated from the mould into the surrounding environment

during solidification is more important.

The total intensity of the heat flow from the mould to the surrounding environment

depends on the difference between the temperature of the outer surface of the mould Tf and

the ambient temperature TOK, cooled surface S and the total effective coefficient of heat sum

which consists of a radiation component and a convective component:

)( radkoncelk

okfcelkokf SdtTTdQ

Accumulation of heat from the metal in the shell is defined based on the heat transfer

between the casting and the mould by convection at the interface of the two environments.

Anytime, the heat flow is proportional to the coefficient of heat transfer between the metal

and the mould and the difference between the Tk (metal) and the Tf (mould) temperatures. A

part of this heat is accumulated in the mould in accordance with the following relationship:

Tcmq ffakf

.

Where T is the temperature change, mf is the weight of the mould element and cf is

the specific heat of the mould.

Heat dissipation by convection which is the easiest part of the calculation is governed

by Newton's law of cooling defining the dependence of heat flow density on the heat transfer

coefficient. The intensity of heat transfer is expressed by the heat transfer coefficient whose

size depends on the properties of the medium, the speed and flow pattern and geometries of

the surfaces involved. The value can be determined by the Nusselt criteria:

CLNu

where LC - is the characteristic dimension (defined by the geometry of the body flown

around)

- thermal conductivity coefficient of the liquid

The intensity of the heat radiation of a body depends on the temperature and the

"radiation ability" of its surface. Real bodies radiate like a gray body, i.e. a body for which we

assume that the relative spectral radiance is constant within the entire wavelength range. If

real bodies do not show too high temperature variations, they behave like gray bodies. Real

values of the total spectral emissivity mainly depend on the material and the surface character

(e.g. quality of machining). Emissivity of ceramic materials is between 0.4 and 0.8.

Emissivity can be experimentally determined using a thermal imaging camera scanning the

surface of the shell with a thermocouple which records the surface temperature required for

the calculation of emissivity.

For real shell configurations, this thermal situation can be only solved using numerical

simulation. To perform the calculation, boundary conditions and the impact of geometrical

configuration of the entire system must be analysed with a sufficient accuracy. Numerical

solution of the heat transport from the liquid metal and from the mould into the surrounding


75

environment requires entering the initial and boundary conditions and thermo-physical

parameters of all components.

Figure 38 shows a diagram of shell annealing and metal casting and solidification.

This diagram shows simplified material thermo-physical data, and the initial and boundary

conditions which are necessary for the calculation of heat transfer. However, many of these

parameters are not included in simulation program databases or they are not sufficiently

verified and therefore must be determined by experiment specifically for the given process.

Fig.38. Diagram of heat transfer during the precise casting process and thermo-physical data

required for numerical calculation


Shell mould

Newton's law of cooling

Nusselt criterion

Simulation programs in foundries

76

6.3 Pressure die casting

Study time: 1 hour


define the basics of the pressure die casting technology

define the areas which are preferably monitored during precision die

casting simulations

Presentation

Pressure die casting is one of the most widely used technologies of casting production.

Pressure die casting is the most important technology for the production of aluminium

castings. The production is based on the injection of molten alloy into the metal mould cavity

at a high speed of 40-60 m/s and solidification under high pressure up to 250 MPa. Under

these conditions it is possible to produce castings of very complicated shapes with wall

thicknesses of about 1-2 mm; under certain conditions and for certain alloys even less than 1

mm. Dimensions of castings are very accurate - for smaller sizes, the 0.3-0.5% accuracy can

be achieved.

Numerical simulation allows the monitoring of pressure die casting in all its phases:

cycling

movement of the piston in the chamber

filling the mould cavity with metal

venting

cooling of castings

additional pressure

Knowledge of the distribution of temperature fields throughout the entire casting cycle

is very important, not only with regard to the mould stress, but also for the design of effective

systems of cooling or tempering channels. Another output from the cycling is also

determining the exact temperature of the mould at the beginning of the cycle to calculate the

mould filling.

Casting quality during the production cycle is significantly affected by the first phase

of pressing. Piston movement and behaviour of the metal in the chamber can predict the

amount of entrapped air in this phase of the process, which may result in a high amount of

defects of certain types. In addition, the simulation of piston movement can be used to

determine the velocity of the metal in the runner, which is one of the key parameters as


77

regards the final quality of castings. At the same time, areas with turbulent filling can be very

easily detected and the proper positioning of overflows can be evaluated.

By changing geometrical parameters or the percentage of the chamber filling it is

possible to avoid adverse situations resulting in various defects (e.g. porosity). Porosity can be

reduced by controlling the process parameters such as the speed of filling, cooling intensity,

amount and type of mould spraying, and the mould venting.

.

Fig.39. The temperature field progress of a pressure die-cast castings

The knowledge of the metal movement in the mould helps to define the critical points

of the casting structure which in real experiments appear as defects in castings. Simulation

can be used to define the optimum position of the inlet and overflows, venting, optimum

speed of filling the mould cavity and verification of the casting temperature of the metal.

Porosity, as already mentioned, is one of the most common defects in castings

produced by high-pressure die casting. Development of porosity, whether caused by

shrinkage due to inadequate replenishment of the liquid metal between the liquidus and

solidus temperatures or by mixing air into the fast flowing melt during filling of the mold, can

be predicted using simulation programs (Fig. 38) and subsequently the parameters which have

a significant impact on the porosity can be adjusted.

This mainly includes the working pressure which is exerted by the piston and

replenishes metal when shrinkage occurs. Filling time and velocity in the runner are also

included. The method and type of coating to protect the mould has a significant impact as

well. Shrinkage of the material is the same at all points, but the porosity caused by shrinkage

only occurs in the thick walls of the casting, i.e. in places which solidify last. This effect can

be solved by controlled cooling and by checking the mould.


78

Fig.40. Speed of filling a model casting

Fig.41. Analysis of micro-porosity for the HPDC technology


79


Runner

Cycling

Shrinkage and shrinkage porosity

Additional pressure


80

7 Simulation programs in foundries

Study time: 3 hours


define ......

describe ...

solve ....

Presentation

7.1 Historical development

Generally, computer simulation can be described as a highly effective tool for the

optimisation of processes using high performance computers. In the 1960s, analogue

computers started to be used for solving certain tasks of unsteady heat and mass transfer -

mould filling and solidification processes were solved using numerical simulations on large

computers, in those times owned by large companies or research institutes.

In the 1980s, when it was possible to involve major European university institutions

specialised in foundry technologies, the first foundry simulation software applications focused

on solidification of castings appeared. In Japan, they were known as Ishikawajima Harima,

Kawasaki Steel, Kawasaki Heavy Industry, Komatu Seisakusho, Kobe Steel, Toyota; in the

USA then Cast Anasys, Marc, and Mitas II emerged.

In Europe, the first simulation programs were developed at the Foundry Institute of

RWTH Aachen in Germany (the software had no name) and in England - the Duct simulation

program. Before 1990, some programs only dealt with the heat transfer, e.g. SOLSTAR from

Foseco. They showed the founders that simulation is not designed solely for scientists and can

provide valuable specific results in practice as well.

After the GIFA exhibition in 1989, the first complete programs appeared on the

market, aimed at the filling of moulds These included, for example, MAGMA-soft, ProCast,

Flow3D and SIMULOR.

Today, the European market offers a wide range of comprehensive foundry simulation

programs which allow users to solve various tasks and are continuously innovated and

upgraded.

.


81

7.2 Overview of simulation programs

Modelling of solidification, which is mostly solved when designing or modifying

technologies, consists in solving the equation of heat transfer using the enthalpy method

associated with models adapted to certain groups of alloys. Deformations of castings are

calculated using thermo-mechanical models. In addition, visualisation of isotherms, solidified

parts, and defects are also included, such as shrinkage and deformation occurring during the

solidification and cooling of castings. Additional modules predict the micro-structure of

spheroidal graphite cast iron, grain size, the occurrence of bubbles in Al alloys, etc.

For these purposes, simulation programs include mathematical processing of various

equations and physical laws:

Navier-Stokes conservation of momentum equation

Fourier differential equation of unsteady heat transfer

The laws of rigid body mechanics during plastic and elastic deformation

The equation for determining the stress and deformation

Transformation and structural diagrams

Implementation of simulation programs in the castings production process is shown in

the diagram in Fig. 42.

Fig.42. Inclusion of simulation programs in the development of technological process


82

Overview of the most frequently used simulation software applications used for the

simulation of casting programs is shown in Fig . 43. In the simulation programs of foundry

processes, computational modules mostly use the finite difference method (FDM) and finite

element method (FEM).

Simulation programs are under a continuous development, their accuracy is increased

and various modules designed for specific issues (prediction of microstructure, etc.) are

upgraded.

Calculations have been shortened from several days to several hours. This has made

the numerical simulation a tool for the dialogue between the participants in the development

of castings and allowed introduction of the simultaneous engineering and RAPID

PROTOTYPING methods in the foundry industry. This increased competitiveness of

foundries which are not mere contractors supplying semi-finished products anymore, and

became direct participants in the product creation process.

Program Country

of origin

Method

of grid

creation

Filling Solidifi

cation

Residual

strain

Structur

e Note

PROCASTT

M USA FEM X X X X 3D

SIMTEC SRN FEM X X X X 3D

MAGMASO

FT

SRN FDM/

FEM

X X X X

NOWAFLO

W

NAVASOLI

D

SWEDE

N

FDM X X - X

PROCASTT

M

SIMULOR

FRANCE FEM/

FDM

X X - - 3D

THEL SRN FDM - X X -

SOLSTAR ENGLA

ND

MM - X - -

CAP USA FEM X X X X 3D

Fig.43. Overview of the most common simulation programs used in the foundry industry.


83

7.3 MAGMASOFT®

The MAGMASOFT® software is the most frequently used simulation system - about

750 installations of which 12 are in the Czech Republic - developed and distributed by

Magma GmbH, Germany.

It is a comprehensive modular simulation program developed at the Technical

University of Aachen in cooperation with MAGMAsoft® GmbH Aachen and the Technical

University of Copenhagen.

It consists of individual modules and it is a highly sophisticated 3D simulation

program which allows one to view the dynamics of melt flow, solidification and cooling of

castings in moulds, fluid flow, heat transfer and residual stresses for all major foundry

processes. Another feature of the program is the calculation of mould erosion both in sand

moulds and permanent moulds for high-pressure die casting. The calculation is based on

reference values which result in erosion if exceeded. Very effective function is the calculation

of positive air pressure generated during filling the mould cavity. Inflowing melt compresses

the air which is inside and can escape via the mould itself or through venting channels. It is a

tool which foundry technologists use to significantly simplify the design and subsequent

optimisation of the inlet and vent systems.

It includes its own CAD interface for geometry creation and preparation of computer

networks. The simulation program works based on the finite difference method. This process

of generating a network is carried out fully automatically and the mesh generation takes

approximately 1 minute.

The user can enter this automatic process to determine the size of each element and

their mutual ratio. The advantage of the FDM method is its speed, automation and accuracy

without the necessity of detailed knowledge of mesh generation.

The mesh generation provides a mesh not only for the casting but also for the inlet and

riser systems, mould, core and cooling channels. Using the differential method, the task is

converted based on the differential operator (usually using the Taylor series) to a differential

equation which is used for solving different bodies with certain limitations - boundary

conditions for the solution of differential equations.

In the MAGMASOFT®-5 software, all steps are performed simultaneously within a

parallel simulation process: it allows the interactive viewing and definition of moulding

process, handling of geometry and simultaneous evaluation of results.

It can be applied for the following processes:

steel, cast iron, aluminium alloys and non-ferrous metals,

for casting in sand and metal moulds,

for gravity casting, low pressure and pressure casting,

precision shell mould casting.


84

.

Fig.44. Description of the basic modules of the MAGMASOFT® software

The following text describes some of the basic modules of the MAGMASOFT®

simulation program and explains their functions.

MAGMASOFTfill is a module which simulates filling of the mould cavity with liquid

metal. It solves the gating system filling, estimates the potential erosion of moulds, calculates

the filling time and other criteria. In addition, it monitors the flow and the occurrence of

turbulent areas, monitors the pressures and temperatures in melt and the velocities of metal

flow in each production phase. Figure 42 provides an example of the simulation of filling

during high pressure die casting (modification of the runner shape). On the left you can see an

incorrect junction of a gating system. Due to the presence of air which is at first entrapped in

this area and subsequently flushed into the casting space, increased porosity can be expected.


85

The same situation is shown in the figure on the right, only it is displayed using the

trace particles which can be used to detect the turbulent nature of the mould cavity filling.

Fig.45. The course of filling during the high pressure casting.

MAGMASOFTbatch is designed to solve the pouring in casting cycles into

permanent moulds (metal moulds). This part describes the distribution of temperature field

and the conditions of flow, cycle times at different criterial conditions, the mould and pouring

temperatures at the moment of opening and at the beginning of a new cycle, and the optimum

opening time.

MAGMASOFThpdc is used for analyses of the high-pressure casting. In this method

of casting, each phase of production process is taken into account, e.g. filling the filling

chamber, the piston movement and filling of the casting itself. The module allows simulation

of any number of cycles, the control of cooling circuits, the application of spray and paint

on the mould, as well as additional pressure during solidification (local squeeze casting).

MAGMASOFTsolid deals with the problems of thermal flow in moulds, taking

account of the temperature-variable properties of melt while providing the information on the

mould filling and porosity. It takes care of solidification times, temperature gradients and

cooling state in each point, the thermal stress of cores and moulds, cooling curves, the

suitability of riser locations, as well as their potential refilling.

MAGMASOFTpost is designed to analyse simulation results. These results are

presented in three-dimensional colour views and describe, for example, the speed and time of

the mould cavity filling, the flow direction vectors of the criterion function for shrinkage,

cooling curves and additional solidification ability of risers.

MAGMASOFTthixo is a module for the simulation of thixotropic casting process.

This method of casting production is currently becoming an alternative to casting and forging.

To simulate this method of casting production, particularly of aluminium and magnesium

alloys, a special equation of motion is used to simulate the filling. Figure 43 shows the course

of filling during the thixotropic casting.

There is a number of applications of stress and deformation simulation in the foundry

production. This mainly includes shape changes, the occurrence of hot cracks, the

development of residual stress, stress in the mould, etc. The MAGMASOFT® simulation

program can predict stress and deformation in castings.

The calculation is based on the temperature field in the casting during filling,

solidification and local cooling, based on the results from the standard MAGMASOFT®


86

software. The results can be presented as the normal stress in the X, Y or Z axes, von Miess

stress, distortion and displacement in each axis. In addition, a criterion for describing the

occurrence of cracks and stress gradients is available for the user. Based on the distribution

of residual stresses (Fig. 44), heavily stressed areas can be predicted as well as a potential

deformation of the casting.

Fig.46. The course of filling during the thixotropic casting

Fig.47. Distribution of residual stresses in a casting


87

7.4 ProCast

It is a professional foundry simulation program developed by UES Software, Inc.,

USA. This system is used for simulation of thermal processes of conduction, convection and

radiation. It allows one to simulate the dynamics of melt flow and heat flow, optimise the

gating system, calculate the stress and deformation of castings, predict metal structure

changes and casting defects during the mould filling and solidification, and determine the

parameters of the production process. It is very compatible with results obtained through

experiment.

This program can also be used to define and establish process conditions for the

production of castings using the gravity casting into both the sand moulds and permanent

moulds, the low-pressure and high-pressure permanent mould casting; it is used to determine

the specific characteristics of the shell mould casting and the investment and evaporative

pattern casting. In addition, it allows simulation of vacuum casting, centrifugal and tilt

casting, continuous casting, monocrystalline casting and other specific technological

processes.

The program includes the Mesh-cast preprocessor which allows the creation of

geometry and the preparation of computational mesh. The geometry from the CAD system

can be transferred in the IGES, Step, STL or PARASOLID formats. Computational meshes

can be transferred from the I-DEAS software. This simulation program is based on the finite

element system and can be used to deal with:

formation of porous areas, shrinkage porosity and cracks,

cold laps, misrun,

stress and deformation of castings,

life of moulds and casting parts,

It is applied in the following processes:

sand mould and metal mould casting,

gravity, low-pressure and high-pressure die casting.

continuous and centrifugal casting,

metal matrix composites,

casting of thixotropic and rheologically complex materials,

squeeze casting,

Fe, Al, Co, Cu, Mg, Ni, Ti, Zn alloys

The company's representative in the Czech Republic is MECAS Plzeň.


88

7.5 PAM CAST / SIMULOR

This simulation program fully solves the Navier-Stokes equations of turbulent metal

flow, as well as the equations of heat balance. The mesh is generated using the finite volume

method (FDM).

It also allows the creation of moulds, their filling, solidification, microstructure and

residual stresses. Plug-in modules are used to determine hardness and deformation. Prediction

of defects can be performed by multiple criteria.

Generally, it is applied for the following processes:

steel, cast iron and aluminium alloys.

casting in sand and metal moulds,

and in precision casting.

The company's representative in the Czech Republic is MECAS Plzeň.

7.6 WINCast /SIMTEC

It is a German simulation program. The basic structure of the WinCast simulation

program consists of modules which provide the required simulation calculation. This program

uses the finite element method for the solution.

The software is able to simulate simple filling of moulds and solidification to predict

defects related to convection. It performs the analysis of solidification and can predict defects

including shrinkage, cracks, segregation and residual stress. The distribution of temperatures

during solidification and/or cooling is calculated not only for points, but also for the entire

volume of the casting. In addition, the distribution of temperatures in the casting mould can

be performed.

It is used in the following processes:

sand mould and permanent metal mould casting,

precision shell mould casting.

Pressure die casting (low-pressure and high-pressure)

lost foam,

semi-solid metal casting,

continuous casting

centrifugal casting

squeeze casting.

7.7 Nova Flow & Solid

These are Swedish simulation programs which are based on the finite difference

method. It consists of a separate module for 3D shape modelling and mesh generation, as well

as modules for heat transfer, flow, solidification and calibration. The model geometry is

created using the CAD files in the STL or DXF formats. Simulations can be monitored

through temperatures, liquid phase and shrinkage, and the determination of solidification

time, flow velocity and visualised 2D or 3D slices in rotation is provided.


89

The calculation can also include the coating and insulation of moulds, locations of

filters, etc. The system is based on a common solution to the flow equations with the

application of incompressible fluid, depending on the Reynolds number, and on the

calculation of friction losses and gravity direction. It also uses the heat transfer equation.

When compared with the above simulation programs, this program is easier to use and is only

intended for informational, i.e. to predict the solidification and cooling of castings.

It can be used for the following processes:

sand moulds with horizontal and vertical parting planes,

precision shell mould casting,

permanent moulds,

casting of steel, cast iron, aluminium alloys and copper alloys.

Key to solutions

90

8 Key to solutions

Here are answers to theoretical questions of each chapter which will test your

knowledge of the studied subjects.

O 1.1. Solidification of metals and alloys has two stages. Solidification begins by crystal

nucleation followed by crystal growth.

O 1.2. The mechanism of solidification (crystallisation) of alloys determines the

microstructure of the alloy and therefore its mechanical properties.

O 1.3. The character of microstructure can be influenced by intervention in the

solidification of castings, inoculation or disturbed crystallisation, or by external

forces (vibration, ultrasound, etc.)

O 1.4. Changes in the free enthalpy G

O 1.5. Crystallisation is caused by the metal or alloy trying to achieve a stable state

when being cooled. From the thermodynamic laws' point of view, a stable state is

defined by a minimum of free enthalpy.

O 1.6. In homogeneous nucleation, nuclei of a new phase are formed directly from the

original phase. It is a "laboratory" case where large supercooling is necessary.

Nuclei resulting from heterogeneous nucleation are formed due to the presence of

foreign inclusions, it is a mechanism of nucleation in real metals and alloys.

O 1.7. Which way of nucleation occurs in real conditions of solidification of metals and

alloys?

O 1.8. It must reach the critical size rkr.

O 1.9. Physical basis of crystallisation from heterogeneous nuclei consists in reducing

the interface tension in the melt - foreign particles - the resulting nucleus system,

therefore the value of the energy required for the formation of active nucleus is

lower.

O 1.10. Segregation, i.e. exsolution, means different concentrations of additive element in

the solid phase due to rapid cooling of castings in real conditions of

crystallisation.

O 1.11. In real conditions the surface area of castings contains randomly oriented

crystals (crystallites). This casting structure is linked with a region of elongated

columnar crystals whose main axes are parallel to the direction of maximum heat

removal from the casting and which have a typical dendritic characteristics. In

the centre of the casting is a region of equiaxed globular (polyhedral) crystals

These types of structure may not always be found in castings.

O 1.12 Based on the solidification progress inside the casting, there are two types of

solidification morphology; a) exogenous and b) endogenous

O 1.13. The thermal axis is the set of points where the crystallisation surfaces meet.

O 1.14. The width of a two-phase zone is affected by the solidification interval of the alloy

Key to solutions

91

(it is defined by the chemical composition of the alloy) and the cooling rate (heat

accumulation of the mould bf)

O 2.1. Modern simulation programs include prediction of the melt during mould filling,

interaction of the metal and mould, casting deformation - stress, and prediction of

the structure and microstructure of the casting. Models include the heat transfer,

rates and methods of moulds, the flow of metal in the mould, solidification

kinetics, formation of structures, models, porosity, segregation in the

solidification interval, calculating also the tension.

O 2.2. The purpose of modelling - simulation is to achieve a forecast with the greatest

possible accuracy, saving time and money in the management, operation,

development and production.

O 3.1. The system means a set of elementary parts, elements, that have specific links

between them.

O 3.2. modelling is based on replacing the studied system with a model, the aim of which

is to obtain information on the original studied dynamic system through

experiment.

O 3.3. Simulation is a research technique based on replacing the investigated dynamic

system with its simulator, performing experiments with the simulator in order to

obtain information about the original dynamic system examined.

O 3.4. The term verification means checking correctness of the model, i.e. to rule out

potential bugs in the program or to make sure an inappropriate numerical method

is not used.

O 3.5. Validity of the model is the model validation based on information we have about

the system being modelled and which we want to obtain through simulation.

Through this step, we are trying to prove that we are actually working with a

model adequate to the modelled system.

O 4.1. Through modelling, dynamic properties of the system and the effects of changes in

boundary conditions of operation of the system can be determined, metallurgical

and other systems optimised, and optimisation of the dimensions and other

technical parameters recommended.

O 4.2. modelling of processes is divided into physical and mathematical modelling.

O 4.3. Physical modelling usually addresses processes running on the actual equipment

and its scale models of real devices and at normal ambient temperatures. To this

end, the theory of physical similarity between two systems is used. In comparison

with mathematical models, physical models define properties of the modelled

system more completely and more reliably. Physical modelling solves tasks in the

substance, while mathematical modelling analyses the structure of the problem.

Moreover, during the construction of physical models it is not necessary to know

the mathematical description of the analysed process. Mathematical modelling

which includes experimental and statistical models and analytical models.

Mathematical modelling is based on a mathematical analogy (similarity) of two

different processes. Physical phenomena of different nature are mathematically

similar when they are described by formally identical (isomorphous) basic

equations.

Key to solutions

92

O 4.4. Systems are geometrically similar if the ratio of the corresponding linear systems

in the model and the product is the same; this ratio is referred to as a constant of

similarity.

O 4.5. It expresses the similarity of velocity fields and acceleration fields. Basically, it is

a balance observed between two geometrically similar systems in which the ratio

of velocity in matching locations of the model and the product is constant and in

both systems the direction of velocity or acceleration is the same.

O 4.6. Geometric similarity between the model and the modelled system.

O 4.7. Similarity of forces between two geometrically similar systems in which the ratio

of forces in the corresponding places and times is constant and their direction of

action identical.

O 4.8. Geometric and kinematic similarity

O 4.9. A flow of molten metal until the onset of solidification follows the basic principles

of fluid mechanics, therefore the continuity equation, Bernoulli's equation, Euler

equation, Navier-Stokes equation, and in terms of solidification of castings also

Fourier differential equation of transient heat

O 4.10. It expresses similarity of two systems (model, modelled system)

O 4.11. A dimensionless parameter has the same value in homologous points in similar

systems, meaning it does not change, but has not a constant value in all points of

the systems.

O 4.12 The basic equation describes a physical system; it is formed by unifying the

complete physical equation which takes into account all the relevant variables

with the conditions of uniqueness

O 4.13. It defines the use of criterion equations where the relevant variables are replaced

by the criteria of similarity which are derived from these relevant variables.

O 4.14. In a model, thermo-physical data are of key importance.

O 4.15. Analytical and numerical. Analytical (explicit) solution consists in finding the

exact solution using analytical mathematical methods (solving systems of

equations, solving tasks tied to extreme etc.), while the numerical (approximate)

solution is used to solve models where a problem can not be solved analytically or

where an analytical solution is difficult and too complex.

O 4.16. Conditions of clarity are the boundary and initial conditions characterising the

given system. There is a rule saying that for each independent variable, we need

as many mutually independent conditions as is the highest order of derivative

occurring in the equations with respect to this variable.

O 4.17. Three - Dirichlet, Neumann, Newton

O 4.18. Finite difference method - FDM and finite element method -FEM.

O 4.19. Breaking down the analysed areas into subareas

computer support of casting and solidification...

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