MODELLING STUDIES ON THE EFFECTS OF CONVECTION DURING

8/14/2019 MODELLING STUDIES ON THE EFFECTS OF CONVECTION DURING

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Chapter 1

Introduction

In industrial applications, solidification is the removal of heat from molten alloy in a

controlled manner to transform it into a solid. Solidification processes are driven by energy

and mass transport, and thus the properties of solidified components like segregation,

structure, porosity, strength etc. are finally controlled by either or both of these phenomena.

In majority of the cases, the transition from liquid to solid takes place in the presence of

convection. Initially the transfer of the molten alloy into a mold under the action of gravity,

usually termed as pouring or filling, is an important source of convection which has a strong

effect on the initial solidification behaviour and the resulting microstructure of the ingot or

casting. Once the initial momentum of pouring is dissipated, natural convection sets in as a

consequence of density differences resulting from variations in temperature or concentration.

These phenomena together are called as thermo-solutal convection[1].

Convection in the bulk liquid is an important parameter in crystal growth from the melt. The

energy and mass transport during solidification can be due to the conduction of heat and

diffusion of solute and, in addition, convection and radiation may be involved [2]. When a

crystal is grown from a high-temperature melt, the temperature field often plays a significantrole in determining the nature and the properties of the crystal. The temperature field in the

melt gives rise to gradients of density, or surface tension, or both. Such gradients generate

fluid motion which may substantially affect the growth process. Thus convection plays a very

important role in solidification of alloys from melt[3,4].

Sources of Convection:


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Temperature gradient affects density and the difference of densities at different parts of the

melt induces buoyancy effect during solidification process. Besides buoyancy, the surface-

tension gradient that results from temperature or concentration gradients on a free surface is

also very important as driving force for convection. Surface tension typically decreases with

increasing temperature. In a free melt surface with a temperature gradient, surface tension

forces drive the flow from regions of low surface tension (hot) to areas of high surface

tension (cold). Surface tension forces are balanced by viscous shear which transfers

momentum to neighbouring liquid layers because of fluid viscosity. Similar to the buoyancy

driven flow, continuity causes the development of a bulk flow in the whole melt volume. This

convective motion is referred to as Marangoni or thermo-capillary convection and occurs if

free melt surfaces exist in the growth configuration[4].

Effects of Convection on Solidification:

Since melt convection may alter the local heat and solutal transfer at the solid/liquid (S/L)

interface, the microstructure is strongly affected by the presence of flow during solidification.

Therefore, understanding of the effects of convection on micro-structural development is

important for controlling microstructure and hence mechanical properties of castings. Someof the effects specifically are listed below[5]--

a) Morphology of solid/liquid interface: Flow plays an important role in the transition

from planar-to-cellular and cellular-to-dendrite interface (also morphology can change

due to diffusion of solute).

b) Microstructure: Flow affects the evolving microstructure through changes in cooling

rate and solute concentration.

c) Macrostructure: Flow has major effect on morphology and morphological transition

(e.g. columnar-to-equiaxed transition).

d) Segregation: Macrosegregation is widely attributed to the convection in the mushy

and superheated regions. It is also influenced by the morphological transition which,

in turn, is influenced by flow.

e) Microsegregation: It is caused primarily by diffusion. However, since both

morphology and solute concentration are affected by flow, microsegregation is

influenced by it as well.


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Chapter 2

Solidification

Solidification and melting are transformation between crystallographic and non-

crystallographic states of alloys. If a liquid is cooled below its equilibrium melting

temperature, Tm , a driving force for solidification will be there. This driving force is nothing

but the difference of free-energies between liquid and solid phase, i.e. GL- GS. Supercooling

at any boundary of the molten liquid brings about super-saturation which acts as driving force

for nucleation.

Alloy Solidification

Mainly, this report focuses on structural applications where alloys are extensively used. On

the other hand pure metals, useful for functional applications, are soft ones. Even

commercially made pure metals contain impurities to change exploit different usefulcharacteristics during solidification. In this context, the solidification of single-phase binary

alloy will be discussed.

Two most important applications of solidification are casting and weld solidification. Most

engineering alloys begin by pouring or cast into a fireproof container or mould. If the as-cast

pieces are permitted to retain their shape afterwards, or are reshaped by machining, they are

called castings. If they are later to be worked, e.g. by rolling, extrusion or forging, the pieces

are called ingots. Ingot structure consists of 3 zones --- Chill zone, Columnar zone, equiaxed

zone[6].

Chill Zone:

It is on outer portion of ingot. Crystals close to mold wall form chill zone. As soon as the

molten metal is poured and it comes in contact with the mould wall, it becomes rapidly

cooled below the liquidus temperature. Many solid nuclei then form on the mould wall and

begin to grow into liquid.


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Fig.1 Different zones in an ingot structure [6].

Columnar zone:

Very soon after pouring the temperature gradient at the mould walls decreases and the

crystals in the chill zone grow dendritically in certain crystallographic direction. Those

crystals perpendicular to the mould walls grow fastest and are able to outgrow less favourably

oriented neighbours. This leads to the formation of the columnar grains all with parallel to the

column axis. As the diameter of these grains increases additional primary dendrite arms

appear by a mechanism by which some tertiary arms grow ahead of their neighbours.

Fig.2 Competitive growth of dendrites soon after pouring[6].

The region between the tips of the dendrites and the point where the last drop of liquid is

solidifying is known as mushy or pasty zone.

Equiaxed Zone:


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Cooling

Cooling

Fig.4 Schematic of an equiaxed solidification system[1].

Still there is no such definite model to calculate the grain generation due to fragmentation of

dendrites in the presence of melt flow, though a lot of experiments have beendone[1,7]. Grain generation rates are used in the following conservation equation for

calculating the local grain density, .

Here, is the death rate of grains due to complete remelting. The term represents the contribution from convection in the melt.

Interfacial Drag:

For isolated small nuclei and globulitic grains, the well-known Stokes' law,


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can be expected to be appropriate. For dendritic grains, there can be flow of liquid through

and around a grain. Thus, the knowledge of the interfacial drag between equiaxed grains and

the melt is important for calculating their relative motion.

Fig.5 A single equiaxed dendritic grain[1].

The situation is further complicated by the presence of other grains. In the limit of packed

grains, all of the flow must be through the dendritic structure of the grains. In this case, a

permeability associated with Darcy's law is typically used to characterize the drag. The most

commonly used expression for representing the permeability function is the Blake-Kozeny

correlation, which is represented as,

Here, the permeability of mushy region is which is used in Darcy term of momentumequations. The permeability coefficient, in the Blakekozeny equation basically dependson microstructural details, namely, primary and secondary dendrite arms spacing. the

permeability constant, (m2) may be evaluated as [7],

Here, the dendritic arm spacing has been denoted as (m).


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Solute Transport:

Fragment transport to open liquid region requires fluid flow by natural convection and/ or

induced stirring. The fluid flow may or may not play some part in fragmentation process.

Because of the density difference due to thermal and solutal gradient, natural convection

occurs within or without the mushy region. In reality, the contribution of solutal gradient has

more prominent effect on convection than that of thermal gradient. It may be assumed that

the interdendritic liquid inside the fine dendritic structure of an equiaxed grain is well mixed.

Let us assume that at a particular local temperature, the liquidus concentration is . On theother hand, the rejected solute is transported from the envelope into the constitutionally

supercooled liquid by the diffusive as well as convective transports. As reported in Rappaz

and Thevoz[8]

, the limit of purely diffusional transport, of an isolated grain during steady

growth, is given by

Here, the liquid mass diffusivity is and the outward velocity of the envelope (speed ofdendrite tip) is . Internal solid fraction of an isolated grain may be expressed as,

Here, the dimensionless solutal supercooling is denoted by. is the concentration of themelt far away from the grain and is the partition coefficient.In the presence of convection, the diffusion length, can be associated with a convective

mass transfer coefficient through

Then, a solute balance (assuming an immobilized interdendritic liquid) leads to

Generally, because convection enhances the solute transport away from the grain,resulting in a higher internal solid fraction compared to diffusion. In the limit of no


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convection, . Note, however, that the dendrite tip speed in the presence ofconvection is much higher than in the diffusion.

Dendritic Tip Growth:

The dendritic tip speed is few times higher than that of pure diffusional growth in melt. There

are literatures available to quantitatively measure the tip speed. But the complicated flow

fields near the multiple tips of an equiaxed grain make it really difficult to come up with

some straight forward model equation[9].

Fig.6 Schematic of dendritic tip growth[9].


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Chapter 3

Convection and Its Effects on Solidification

Till now, only the solidification (specially the effect of dendritic expansion) has been studied

in this report. In this section, the effect of different types of convection and their effects will

be discussed and thereby will represent a promising opportunity to obtain some concepts to

improve crystal quality. As a relevant example, the directional solidification of a binary alloy

in an open-boat configuration (Sampath & Zabaras 2000)[4]will be discussed in this section.

Fig.7Schematic of the binary alloy solidification problem in an open-boat configuration[4].

For this particular situation, solidification process takes a very interesting turn. Here, along

with the diffusional mixing in the liquid phase, a strong recirculation force also comes into

play as it is shown in Fig.7. A comparative study between density- and surface tension-driven

convection has performed under different levels of gravity as reported in [4]. So, basically this

particular problem has been presented as an example of directional solidification process

where the combined effect of buoyancy as well as surface tension forces are simultaneously

acting on the system.

As reported in Sampath & Zabaras (2000), a two-dimensional, rectangular mold is

considered filled with a dilute, incompressible binary alloy which is initially liquid and

uniform in temperature and composition (Fig. 7). The top boundary of the domain is kept

free and subject to temperature induced surface-tension gradients. At time t =0

+

, thetemperature of the left vertical boundary, sis instantaneously dropped and maintained below


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the freezing temperature corresponding to the bulk concentration of the melt, so that

solidification of the alloy immediately commences at this cold boundary. The motion of the

melt is determined by the combined action of buoyancy, and surface-tension forces.

The melt is modelled as a Boussinesq fluid and the transient NavierStokes equations are

solved simultaneously with the transient heat and solute transport equations.

As it is described earlier that driving force for convective flows exist in common

configurations of crystal growth from melt ---

1. Density gradient within the fluid under a gravitational field. It leads to free/natural

convection in the melt.

2. Pressure gradients created by any external force leading to forced convection.

3. Marangoni/thermocapillary effects due to surface tension gradients in the melt.

In this context, the effect of forced convection will not be discussed as it is assumed that there

is no such external force present to produce it.

In the present problem, the solid and liquid region are denoted bysandl respectively,

and the solidliquid interface as i. The region lhas a boundary l which consists of i, ol

(the mold wall on the liquid side), bl(the bottom boundary of the liquid domain), and tl(the

top boundary of the liquid domain). Similarly shas boundary s, which consists of i, os,

bs , and ts. In the solidification system considered in this work, the thermal and physical

properties are constant in each phase. Melt flow is also assumed to be a laminar flow induced

by thermal, solutal, and surface tension. It also may be assumed that temperature difference

in the liquid is so small that the density difference in it is also very negligible. Still the

buoyancy driven convection is present and thus Boussinesq approximation can be applied.

The free-surface (as shown in Fig.7) deformation is negligible. Also GibbsThompson effect

may be negligible which means capillary undercooling due to the curvature of the phase

boundary has no such effect on the present problem. A macroscopically stable solidliquid

interface is assumed to exist between the solid and the liquid regions.


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The above assumptions with the exception of the last three are generally valid for

solidification systems with dilute concentration levels, moderate temperature differences, and

a Newtonian melt. The assumption that the melt free surface deformation is negligible is

strictly valid only at low capillary numbers. This is generally true for metals and

semiconductor melts [10]. The GibbsThompson effect is neglected as it is usually important

only on very small scales [11,12]. Finally, the assumption that a sharp solidliquid interface

exists between the solid and liquid domains is quite debatable. Once morphological

instability occurs, the perturbations continue to grow to form a two-phase mushy zone in

which dendrites are bathed in interstitial melt. Many theoretical models for simulating

dendritic solidification processes incorporating a mushy model have been proposed in the

recent years [9,13]. In this report, however, the simplified model of a sharp solidliquid

interface is adopted to allow us to concentrate on examining the complex interactions of heat,

mass, and momentum transport in the solidification system. A number of computational

studies have been conducted using the sharp interface model, and the results were found to be

on an excellent qualitative agreement with experiments [14-17].

The governing equations for the binary alloy solidification system are now introduced. Let be a characteristic length of the domain, the density, k the thermal conductivity,

( ) the thermal diffusivity, Dthe solute diffusivity, ethe electrical conductivity, andthe kinematic viscosity of the liquid melt. The characteristic scale for time is taken as and for velocity as . The temperature is defined as

,

where and are the temperature, reference temperature, and reference temperaturedrop, respectively.

Likewise, the dimensionless concentration field is defined as , where and are the concentration, reference concentration, and reference concentration drop, respectively.

The basic equations used in the simulation of the melt flow are the incompressible Navier

Stokes equations. The Boussinesq approximation is also used for defining buoyancy. The

other main equations governing the fluid flow in the liquid domain are the energy equation

and the solute transport equation.


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These equations are written as

for incompressible, steady flow,

where the governing dimensionless groups are the Prandtl number (=), the Lewis number( ), the thermal Rayleigh number (

), the solutal Rayleigh number (

), and the Hartmann number ( ), where g is the gravity constant, is the

thermal coefficient of expansion, and is the solutal coefficient of expansion. Here, denotes unit vectors in the directions of the gravity vectors. Heat transfer in the solid is by

conduction and is written as

where is the ratio of the thermal diffusivities.Bond number,

.

where Marangoni number. A detailed discussion on these numbers are presented innext section.


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A no-slip and no-penetration boundary condition on the velocity field is imposed on all the

liquid boundaries other than the upper free surface (Fig.7). The surface tension on the free

surface is approximated as

where as

The variation of the surface tension with solute concentration is neglected as only minute

amounts of solute are considered and hence no appreciable change in surface tension occurs

as a result of concentration variation [13,18]. The hydrodynamic boundary condition on the free

surface tl is expressed as where is a tangent vector to the free surface, and Marangoni number,

Hence, to understand the individual contributions of density difference and Marangoni effect

to convective flow is really important. But also it is really difficult to understand the

individual or rather the effect of surface tension in convection. On earth convection due to

density difference is dominant over that because of surface tension. So, to separate out

Marangoni effect, simulations are done under reduced gravity. Such simulations clearly

separately show the exact contribution of surface tension to convective flow. Now, a more

complete governing equation capturing the physics of both the effects of density and surface

tension differences are obtained. So, if the experiments are done under gravity we can now

use our prior knowledge to strike a balance between these density driven convection and

Marangoni effect[19-22].

Thus, to understand the relative influence of thermocapillary convection and buoyancy driven

convection flow on solidification process, a particular dimensionless number is used which is

called as Marangoni number.


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Chapter 4

Results & Discussion

As reported in Sampath & Zabaras (2000), the simulations were performed with Antimony

doped Germanium (SbGe alloy) and the different results are presented and discussed in this

section.

It can be observed that initially the thermal gradient in the melt leads to density driven

convection in bulk liquid region and surface tension driven convection on the top free top

surface. As shown in Fig.8the fluidvelocities are maximum at the top free surface in regionsclose to the interface i. Continuity of the fluid flow slowly leads to counter-clockwise

circulation of the melt filling the entire cavity[4]. As the solidification process proceeds

further, the interface i starts to curve, with more solid volume formed at the bottom

compared to the top part of the cavity.


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lower part of the cavity at this time. As the solidification proceeds further, the strength of this

recirculating convection flow slowly increases, along with a steady increase in the size of the

convection cell[4].

Fig.10Calculated contours of stream function, solute concentration, and temperature fields at

times =3 and =6 for the solidification of SbGe under reduced gravity (g=10-5

gearth )conditions and zero magnetic field[4].

Around = 2, the main recirculating cell fills almost the entire cavity. At the same time, a

secondary cell pattern forms at the right end of the cavity. After around = 3, there is almost

no change in the structure of the main cell, even though its strength steadily increases with

time as shown in Figs.10 & 11. This complex evolution of the melt flow has significant

impact on various solidification parameters.


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Chapter 5

Conclusion

The role of convection during solidification of binary alloys was investigated with the help of

mathematical models. Firstly, the basic theory of alloy solidification is discussed. Later, to

assess the comparative effects of thermocapillary-, and buoyancy-driven convection, a

systematic series of simulations were performed under different gravity levels.

The reported calculations demonstrate that thermocapillary convection plays an important

role in the solidification process and hence Marangoni effect prominently appears up. Under

low-gravity conditions and in the absence of any external magnetic field, the melt flow

develops at the free surface and slowly diffuses into the liquid. This particular phenomena

leads to the local accumulation of solute and formation of various high solute concentration

spots.

This report not only reveals the effect of buoyancy- or surface tension-driven convections in

liquid alloy but also makes an attempt to establish the advantage of using Bond number of

convection current in the liquid region. Thus, this report points to a very basic footstep to

control microstructure and properties of cast products by controlling the convection currents

in the liquid region.


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MODELLING STUDIES ON THE EFFECTS OF CONVECTION DURING