NUMERICAL SIMULATION OF SOLIDIFICATION AND MELTING PROBLEMS USING ANSYS FLUENT 16.2.pdf

7/26/2019 NUMERICAL SIMULATION OF SOLIDIFICATION AND MELTING PROBLEMS USING ANSYS FLUENT 16.2.pdf

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NUMERICAL SIMULATION OF SOLIDIFICATION AND

MELTING PROBLEMS USING ANSYS FLUENT 16.2

BY

SHUBHAM PAUL (10300712142)

DEBORIT DE BISWAS(10300712109)

WASIM SAJJAD(10300712153)

RAKESH KUMAR JHA(10300612037)

Under the Supervision of

Astt. Prof. Debraj Das

DEPARTMENT OF MECHANICAL ENGINEERING

HALDIA INSTITUTE OF TECHNOLOGY

HALDIA-721657

MAY, 2016


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CERTIFICATE

This is to certify that the work contained in the thesis entitled Numerical Simulation of

Solidification and Melting Problems using Ansys Fluent 16.2 by Shubham Paul

(University Roll no. 10300712142), Deborit De Biswas (University Roll no .

10300712109), Wasim Sajjad (University Roll no. 10300712153), Rakesh Kumar Jha

(University Roll no. 10300612037) of the Department of Mechanical Engineering, Haldia

Institute of Technology in partial fulfillment of the requirements for the award of Bachelor

of Technology Degree in Mechanical Engineering during the academic session 2012-2016

is a bonafide record of thesis work carried out by them under my supervision and guidance.

Neither of this report nor any part of it has been submitted for any degree or any academic

award elsewhere.

.. .

Counter signed by Head of the Department Mr. Debraj Das

(Thesis Advisor)


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Acknowledgement

We would like to take this golden opportunity to convey our sincere gratitude to Asst. Prof.Debraj Das who helped us in carrying our project on COMPUTATIONAL FLUID

DYNAMICSand provided useful guidance without which it would be really tough to complete

this project. He was there in each and every stage to assist and motivate us, so that we could

come up with a good work, and due to his faith and trust upon us, we were able to do this project

work.

This project also made us to know about the difference scope of CFD and the different governing

equations and its link with physical problems of solidification and melting of a material.

At last we would like to thank our Head of Department Prof. Tarun Kanti Janawho has given

us this opportunity to work with Sir. Debraj Das, and get an experience of his expertise in CFD.


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Abstract

In this study, basically we have dealt with Solidification and Melting problem, which is a moving

boundary problem in which we track the solid- liquid interface which moves with time. Natural

Convection and Conduction are the mechanism behind the physics of these problems. We have

solved Navier-strokes equation along with continuity and energy equation, both in solid and

liquid region using structured grid. In order to make zero velocity condition in solid domain

special care has been taken. We have used enthalpy method to track the solid-liquid interface

with respect to time. A fully coupled implicit method is used to solve the momentum and energy

equation. A diffusion phase change, isothermal with convection along with continuous casting

problem are present in the present study, and is validated with analytical and numerical results

available. First, the two and three dimensional diffusion problem has been solved followed bygallium melting and mushy zone problem. Lastly, application problem on continuous casting has

been solved and verified.


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Contents

1 Introduction 2

1.1 Methods needed for solving phase change problems 2

1.2

Literature Survey 4

1.3

Objectives 7

1.4 Thesis Organization 7

2 Mathematical Modeling and Finite Volume Method 82.1

Assumptions 8

2.2

Governing Equations 9

2.2.1 Continuity Equation 9

2.2.2 Momentum Equation 9

2.2.3 Energy Equation 10

2.3

Initial and Boundary Conditions 11

2.3.1 Initial Conditions 11

2.3.2 Boundary Conditions 11

3 Results and Discussion 12

3.1

Diffusion Problem (Isothermal Case) 123.1.1 Two Dimensional Problem 12

3.1.2 Three Dimensional Problem 15

3.2 Isothermal Phase Change With And Without Convection 16

3.2.1 Gallium Melting with Convection 16

3.2.2

Gallium Melting without Convection(Diffusion) 19

3.3

Mushy Zone Problem 21

3.3.1 Two Dimensional Problem 21

3.4

Practical Application(Continuous Casting Process) 254 Conclusions And Scope For Future Work 31

4.1 Conclusions 31

4.2 Scope For Future Work 31

5 References 32


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List of Figures

3.1 Square Cavity Problem without Convection 12

3.2 Square Cavity Problem without Convection 13

3.3 Temperature distribution for square cavity problem (Case-1)

(a)10 sec; (b)20 sec; (c)30 sec. 14

3.4 Position of interface in two dimensional problem (Case 2)

(a) 41 sec; (b) 61sec; (c) 81sec. 14

3.5 Position of Interface (a) t= 0.60sec, (b) t = 0.75sec atz = 2 plane,

(c) temperature contours at t= 0.75 sec atz 15

3.6 Melting of Gallium Problem 17

3.7 Streamlines for Gallium melting (a) 6 min, (b) 9 min. 17

3.8 Temperature Contours for Gallium Melting (a) 6 min, (b) 9 min 18

3.9 Interface position at different times 18

3.10 Melting of Gallium Problem 19

3.11 Streamlines for Gallium melting (a) 6 min, (b) 9 min. 20

3.12 Temperature Contours for Gallium Melting (a) 6 min, (b) 9 min 20

3.13 Mushy region two dimensional problem 22

3.14 Vector plot and mushy region for =0.1 (a)t=100 sec,(b)t=600 sec, (c)t=1000 sec 23

3.15 Comparision of (a) uvelocity at t= 500 sec,

(b) solidus and liquidus line att=1000 23

3.16 Temperature contours for two dimensional mushy region problem

(a)t=600 sec, (b) t=1000 sec. 24

3.17 Solidification in Czochralski Model 26

3.18 Shows the temperature contours for steady conduction solution 26

3.19 Shows Contours of Static Temperature (Mushy Zone) in steady State 27

3.20 Shows the Static temperature contour in transient state 27

3.21 Contours for Stream function at t = 0.2 sec. 28

3.22 Contours for liquid fraction 28

3.23 Contours of temperature t = 5 sec 29

3.24 Stream function Contours at t=5 sec 29

3.25 Contours of liquid fraction at t= 5sec 30


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Nomenclature

A Porosity function for the momentum equations (kg/m3s)

b Constant

C Constant

e Total Enthalpy (J/kg)

g Acceleration due to gravity (m/s2)

k Thermal conductivity (W/mK)

p Pressure (N/m2)

t Time (s)

u, v, w Velocity Component in x, y, z directions, respectively

cp Specific heat at constant pressure (J/kgK)

eT Sensible Enthalpy (J/kg)

eL Latent Enthalpy (J/kg)

fl Liquid fraction

Greek Letters

Molecular Viscosity (kg/m-s)

Diffusion Coefficient of the variable

Latent heat (J/kg)


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Chapter 1

1.Introduction

Now-a-days, the phenomenon of solidification and melting is of great importance in basic

manufacturing processes like casting, welding etc. They have a great impact on many industrial

applications. In earlier days, only analytical solutions were available, which did not give a clear idea

about the process. Moreover, some effects (like convection) were also neglected in those days. So,

implementation of numerical techniques for this kind of problems gathers attention for both present

and future research. Solidification and melting problems are phase change problems, in which a solid-

liquid interface is moving with time and it has to be observed and tracked. One extra condition is

required for solving general governing equations of this kind of problems. This condition is called

Stefan condition and has to be applied at the solid-liquid interface. The Navier-Stokes equations

coupled with the energy equation are solved in the problem domain.

The problems can be solved numerically using computational fluid dynamics (CFD). In the present

study, finite volume method (FVM) is used with structured meshes which can be easily applied in any

arbitrary geometry. Ansys Fluent has been used as a tool to implement CFD, in the following thesis.

1.1 Methods needed for solving phase change problems

There are many methods for solving the solidification and melting problems. As interface moves with

time, they are classified according to the choice of domain.

1. Variable domain method: Here ,in this method, the governing equations are solved separately

in both domains. Here the domain changes with time because the interface moves with time.

For this reason it is called variable domain method. The Stefans condition is applied to track

the interface. So, this method requires adaptive grid generation and we have to track the

interface. Two separate sets of equation for solid and liquid are required.

2.

Fixed domain method: Here, the domain does not changes with time. Governing equations areto be solved in the domain. The main disadvantage regarding this method is that it sometimes

breaks down when interface moves a distance larger than a space increment in a time step.

However, it can be easily solved using variable domain method. Again for solving multi-

dimensional problems, variable domain method is not applicable.


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Isothermal phase change: In this case, phase change occurs at a distinct temperature,

enthalpy change is a steep change at melting temperature. This happens in case of pure metal i.e.,

Tin, Gallium etc.

Mushy region phase change: In this case, phase change occurs over a temperature range i.e.,

enthalpy becomes a continuous function of temperature. These problems are referred to mushy

region phase change problems. The relationship between enthalpy and temperature can be any

type linear, exponential. Here only linear relationship is considered. Binary alloys and all

mixtures follow this relationship.

1.2 Literature Survey

Earlier work related to solidification and melting problems is based on diffusion problems only

and convection effects were not so dominant. A brief review regarding the modeling of

solidification and melting problems can be found in Basu et al. [4] and Hu et al. [1]. Basu et al.

[4] have described different types of methods such as fixed domain method, variable domain

method for solving solidification and melting problems. They have formulated the governing

equations for convection-diffusion phase change problems (isothermal as well as mushy region

phase change case). Hu et al. [1] have formulated the governing equations through stream-

function-vorticity formulation as well as primitive variable formulation.

Lazaridis [5] solved multi-dimensional diffusion problems by directly applying Stefan

condition coupled with the energy equation. They solved four kind of diffusion problems. The

discretization scheme for the cells surrounding the interface is different from that for the interior

cells. They used both explicit and implicit time integration scheme. Voller and Cross [6] solved

moving boundary problems using enthalpy methods. They used finite difference scheme for

spatial and for time discretization both implicit and explicit methods are used. They solved two

region problem and two-dimensional problem and compared the result with analytical and

numerical result. Voller [7] developed implicit enthalpy formulation for binary alloysolidification without taking convection into account and used node jumping scheme for tracking

solid-liquid interface. Crowley [8] extended multidimensional Stefan problems and he solved

solidification of a square cylinder of fluid using enthalpy method when surface temperature is

lowered at a constant rate.


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Gau and Viskanta [9] first took the natural convection phenomenon in solidification and

melting problems. They conducted an experiment for studying the buoyancy-induced flow in the

melt and its effect on the solid-liquid interface position and heat transfer rate during the process

of melt-ing and solidification of a pure metal (Gallium) from a vertical wall. They compared the

solution with Neumann problem and concluded that convection effect can be neglected during

phase change problems. Morgan [10] solved phase change problems taking convection into

account. He used ex-plicit finite element method to solve freezing problem in a thermal cavity.

The basic enthalpy formulation of the governing equation was done by Voller et al. [11]. The

enthalpy formulation is a weak solution method. They divided total enthalpy into sensible and

latent enthalpy. They derived an equation for sensible enthalpy, in which latent enthalpy

appeared as a source term. They solved the equation for sensible enthalpy and from that they

calculated temperature. They used FVM for discretization. They solved a problem considering

the effect of natural convection on isothermal solidification in a square cavity. They used

different technique to make the velocity in solid region zero. They used variable viscosity

method, Darcy source based method and switch-off technique as techniques for making zero

velocity in solid domain. This approach was called enthalpy-porosity technique. Voller and

Prakash [12] modelled a methodology for mushy region phase change problem by taking

convection into account. They used enthalpy-porosity tech-nique as mentioned in Voller et al.

[11] for formulation of governing equations. In mushy region, fluid velocity is not zero and

therefore mushy region contributes to some convection, they assumed that in mushy region flow

occurs through a porous media. They defined permeability to model the flow and they took

same governing equation which relates fluid velocity and pressure, derived from the Darcy law.

= () (1.1)

whereis the permeability of the porous medium. Voller et al. [11] neglected convective latent

enthalpy source term for isothermal phase change case. Voller and Prakash [12] did not neglect

the convective term of latent enthalpy source term as it is not zero in case of mushy region phasechange problem. They derived general formulae for both temporal and convective latent enthalpy

source term. Brent et al. [13] applied the formulation proposed by Voller and Prakash [12], to the

problem of the melting of Gallium in a rectangular cavity. They considered isothermal case and

convection was taken into account. They plotted isotherms and streamlines at different times and

compared their results with the experimental results obtained by Gau and Viskanta [9]. Wolff et


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al. [14] solved problem regarding the solidification of Tin in a square cavity by using numerical

as well as experimentally.

The two sides of the cavity were at a fixed temperature and remaining two were insulated. At last

they compared the numerical result with experimental result. For numerical technique they used

enthalpy method. Rady and Mohanty [15] used enthalpy-porosity technique to solve melting ofGallium .

They validated their result with Wolf et al. [14]. They plotted isotherms and streamlines at

different times in case of melting of Gallium problem. They also plotted the interface position at

different times. Stella and Giangi [16] studied the melting of pure Gallium in a bi-dimensional

rectangular cavity with aspect ratio 1.4. They plotted solid-liquid interface and streamlines at

different times and shown a multi-cellular flow structure built in the process of melting.

Redy et al. [17] studied about the effects of liquid superheat during solidification of pure

metals. They also used the enthalpy-porosity technique. They obtainedsteady state very early forhigher Rayleigh numbers. They plotted Nusselt number variations and temperature profiles for

different Rayleigh numbers. Ghasemi and Molki [18] studied isothermal melting of a pure metal

enclosed in a square cavity having Drichlet boundary conditions in each side. They continued their

computations for Rayleigh number 0 to 108and Archimedes number 0 10

7. They plotted liquid

fraction variation with time, falling velocity of solid phase and shape of the solid-liquid interface.

They found that for low Rayleigh and Archimedes number, both melting rate and solid velocity are

low and melting is almost symmetrical. Melting rate enhances with the higher value of Rayleigh and

Archimedes number.

Gong and Mujumder [19] studied melting of a pure phase change material in a rectangular

con-tainer heated from below. They used Streamline Upwind/Petrov Galerkin finite element

method in combination with fixed grid primitive variable method. Flow patterns for different

Rayleigh numbers were used. They also studied the instability of free convection flow at higher

Rayleigh numbers.

Bertrand et al. [20] reviewed the methods to solve the solidification problems and

compared the results. They gave the results for high as well as low Prandtl number fluids. Hwang

et al. [21] considered the effect of density variation with phase change when tin solidifies in a

square cavity. They used multi-domain method to cope up with abnormal variations of frontposition due to shrinkage.


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1.3 Objectives

The objective is to solve Three and Two Dimensional Solidification and Melting Problem

using Ansys Fluent 16.2 for both Isothermal and mushy region phase change and validate

the simulation results with numerical, experimental and analytical solutions available in

the literature.

1.4 Thesis Organization

A brief introduction along with literature review is presented in chapter 1. Mathematical

modeling, Governing equations and initial and boundary conditions are described in

chapter 2. Problem solving using Ansys Fluent on Solidification and Melting is shown in

chapter 3. At last Conclusion and scope for future works are listed.


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Chapter 2

2.Mathematical Modeling and Finite Volume Method

Nowa-days, Navier-Stokes equations and energy equation in solidification and melting

problems are solved using the fixed domain method. Because of versatility of Fixed domain

enthalpy method ,it can be used for both isothermal and mushy region phase change problems. In

this case, as the position of the interface is obtained as part of the solution, explicit information

about the interface is not required. While solving Navier-Stokes equation in the solid domain,

attention must be taken to make zero velocity condition in that domain. Therefore, the fixed

domain enthalpy method demands some techniques to do that, which is described in the next. As

convective effect is not neglected so the Navier-Stokes equations and the energy equation are

coupled in these problems . In the present formulation, the governing equations have been

considered in Cartesian coordinates system.

2.1 Assumptions

1.The flow is considered to be incompressible, Newtonian and laminar.

2. Properties like thermal conductivity, specific heat are assumed to vary linearly with liquid

fraction.

3.The density variation due to phase change is neglected for closed domain problems (like

square cavity problem). The density variation due to temperature in the liquid domain is

incorporated through Boussinesq approximation. Variable density formulation is to be used in

case of external flow. However, the variable density formulation cannot handle shrinkage effect

during solidification. This needs some special treatment [21].

4. Species transport equation is not solved, so solute buoyancy is not included. Only thermal

buoyancy is considered in the present study.

5.

Viscous dissipation effect is neglected.


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2.2 Governing Equations

The governing equations are as follows based on above assumption are written below.

2.2.1 Continuity Equation

. () = 0 (.)2.2.2 Momentum Equation

The Navier-Strokes equation (in vector form) for laminar, incompressible flow of

Newtonian fluid can be written as follows

() + . = p + . + (.)To make velocities equal to zero in the solid domain, a large negative source term is added to the

above equation. The source term becomes zero when it is liquid domain. So, the equation then

becomes

() t + . = p + . + + A (.)The second source term takes very high value for making the velocities very close to zero in the

solid domain and in the liquid domain, it is simply zero. The equation for A is [13]

= (1)23+ (.)Where flis the liquid fraction ,which is defined as the ratio of volumeof the liquid present in anyparticularcell to the total volume of the cell.

= (.)C and b arre prescribed constants. The equation for A makes the momentum equation to

follow the Carman-Kozeny equation in the mushy region. In mushy region both solid and liquidphase are present and thereforefl always takes the value like 0


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2.2.3 Energy Equation

The general form of energy equation after neglecting viscous dissipation term is

Eqn 2.6

The direct form of enthalpy equation is used in the present work. Enthalpy is split into two parts

i.e.,

= + (.)Where eT is sensible enthalpy and eLis latent enthalpy per unit mass.

= (.)eL=0 in the solid region, eL=L in the liquid region and eL varies between 0 and L for the cells

undergoing phase change. Substituting ein the energy equation,( + ) + . + = . (.)After simplifying the equation becomes

() + . = . () . (.)

Now, as the right hand side of the above equation is in terms of temperature, its contribution tothe diagonal term coefficient is zero. So the equation becomes similar to one of pure convective

equation with the source term, which is less stable during numerical solution. To improve

convergence rate and stability of the above equation, temperature in diffusion is replaced by eT.Then the equation becomes

() + . = .

. (.)

Latent enthalpy term in the right hand side can be considered as source terms. Detailed updation

procedure can be found in [12].


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2.3 Initial and Boundary Conditions

2.3.1 Initial Conditions

Initial conditions combined with the boundary conditions will determine whether the given

problem is a phase change problem or not. As phase change problems are unsteady problems, the

initial conditions play a major role in the solution. For solidification, initially some part of the

domain has to be liquid. According to the types of problems parameters are to be initialized. While

solving energy equation, all initial and boundary conditions have to be in terms of sensible

enthalpy.

2.3.2 Boundary Conditions

Boundary conditions needed for the solidification and melting problems can be Dirichlet, Neu-mann or Robin. The boundary condition is to be implemented as follows. For Neumann boundary

condition is implemented as,

= (.)

For solid walls no-slip boundary conditions are used. For pressure, the homogeneous Neumann

boundary condition is used for velocity specified boundaries. For other boundaries, appropriate

boundary conditions should be specified to the physics of the problem.


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Chapter 3

3. Results and Discussion

In this chapter, problems related to solidification and melting are discussed and verified with

the numerical and analytical solution. Firstly, diffusion phase change problems without convection

effect (Isothermal case only) and then both isothermal and mushy region convection-diffusion

phase change problems are discussed. For solving, phase change problems with convection, a

suitable source term are added in the momentum equations to get zero velocity in the solid domain.

The sensible enthalpy form of the energy equation is solved with the appropriate source terms.

Lastly, a problem on continuous casting has been also discussed in the given thesis, which portrays

the application part of this solidification and melting problem.

3.1 Diffusion Problem

We have solved two benchmark problems on solidification and melting to corroborate thesimulation done through Ansys Fluent, with the numerical and Analytical solution from the

literature. The first question being a 2D problem and the other next one is the 3D problem. Efforts

have been made to simulate and bring the result in accordance with the literature. The propertiestaken in such a way that they matches with some non-dimensional quantities (e.g. Stefan number

or some parameter defined in the literature).

3.1.1 Two- dimensional problem

This is a two dimensional problem in which solid is melted in a square cavity having same wall

temperature. There are two cases for solving this problem.

Case 1:-Cavity wall temperature is 1oC.

Figure 3.1:- Square Cavity Problem without Convection


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Case 2:- Cavity wall temperature is 0.5oC

Figure 3.2:- Square Cavity Problem without Convection

For solving this problem we have to take a square cavity having 1x1 dimension. Here four

interfaces are formed i.e. four solid walls and these are joined to form a single interface. In both

cases, initially the solid is kept inside the cavity at its melting point i.e. Tm= 00C. After that the

temperature of all the boundary walls are increased suddenly. Different physical properties taken

for this square cavity problem are as follows:-

Table 3.1:-Physical properties taken for square cavity problem

k

(W/m-K)

Cp

(J/kg-K)

(kgm-3

)

(J/kg)

(kg/m-s)

(1/K)

Solid 1.0 100.0 1.0 0.0 - -

Liquid 1.0 100.0 1.0 1000.0 0.1 0.01

Now for computation purpose, we have to choose a 81 x 81 grid mesh. And the above givenphysical properties chosen such that these match with non-dimensional parameters like Ra = 10

4,

St=0.1 ,Pr =10.


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(a) (b)

(c) (d)

Figure 3.3:- Temperature distribution for square cavity problem (Case-1)(a)10 sec; (b)20 sec; (c)30 sec.

(d)40 sec

(a) (b) (c)

Figure 3.4:- Position of interface in two dimensional problem (Case 2) (a) 41 sec; (b) 61sec; (c) 81sec


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The solution graph shows interface position at different time inside the cavity. It also shows that

Temperature/ Contour lines are axis-symmetry as the boundary conditions of each faces is same

and the geometry is symmetric. Here the intensity of convection is less as Raleigh number is less

I.e Ra =104,

and as we know that as Rayleigh number increases, the tendency of pushing the solid

increases. And here, Rayleigh number is less so liquid does not have enough potential to push the

solid. So in this case conduction i.e diffusion of heat dominants over natural convection asRayleigh number is less. so , convection is neglected in this problem .

3.1.2 Three-dimensional problem

As we have discussed about the 2-dimensional solidification melting problem previously,

now we are getting interest to study about the simulation of the solidification-melting problem of a

3-dimensional cavity having dimensions (4x4x4). This benchmark problem is taken from the thesis

Lazaridis [5] .Initially in this problem the liquid metal having melting point 00C is kept in the

cavity. Suddenly, the temperature of the of the left and bottom wall is reduced to -3.240C that

means Dirichilet condition is applied and the other four boundaries having Neumann boundaryconditions which means they are adiabatic in nature. Now we will track the position of the liquid

solid interface at a constant plane that is Z=2 at different times and will compare with the thesis of

Lazaridis [5]. During the simulation a time step of .01 is chosen. The properties of the metal are

taken by using some non-dimensional number according to Lazaridis [5]. The properties are given

in the Table 3.2.

Table 3.2:- Properties for 3D problem

k Cp

(W/m-K) (J/kg-K) (Kgm-3

) (J/kg)

Solid 1 1 1 0

Liquid 1 1 1 5

(a) (b) (c)

Figure 3.5-: Position of Interface (a) t= 0.60sec, (b) t = 0.75sec atz = 2 plane, (c) temperature

contours at t= 0.75 sec atz= 2 plane.


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Figure 3.5(a) and (b) is the comparison of the interface of our simulation with the thesis of

Lazaridis [5] at different timings. Figure 3.5(c) is showing the temperature contours at Z=2 and

t=0.75 sec. Here the effects of conduction are much higher than the effect of convection so we

consider only the effect of conduction.

Here we observe that the wall having lower temperature converted to solid very quickly then theposition interface progress in such a way showed the Figure 3.5. The interface is looking like a

parabola.

Table 3.3-:Physical Properties of Gallium

k Cp

(W/m-K) (J/kg-K) (Kgm-3

) (J/kg) (kg/m-s) (1/K)

Solid 32.0 381.5 6095.0 0.0 - -

Liquid 32.0 381.5 6095.0 80160.0 1.81x10-3 1.2x10-4

3.2 Isothermal phase change with and without convection

The phase change problems solved till now, only deals with diffusion. Since, the research says

that convection effect cannot be neglected [9], therefore the problems now dealt with are solved

with convection effect considered and variation obtained is studied and analyzed.

3.2.1 Gallium Melting with Convection

As we know Gallium melting, being a benchmark problem for isothermal phase change problems,convection effect was considered and the simulation was validated using Ansys Fluent.

A problem from Brent et al. [13] is taken to evaluate the simulation obtained. The problem is

defines as stated below:

A rectangular cavity of 0.0889 m in length and 0.0635 m in height is taken in which pure solid

Gallium is initially kept at 28.3C. Suddenly, left wall temperature is increased to 38

C which is

higher than the melting point temperature (Tm=29C) of Gallium and the right wall is kept at the

initial temperature of the solid Gallium. Other two boundaries are insulated. A 42 x 32 grid is

chosen for the simulation. Figure 3.6 shows the computation domain with the necessary boundaryconditions. The physical properties are taken from Brent et al [13]. The properties are shown in the

table 3.3. For the present simulation,A=106

and b=0.001 are taken. A time step of 0.01 is used.


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Figure 3.6: Melting of Gallium Problem

(a) (b)

Figure 3.7: Streamlines for Gallium melting (a) 6 min, (b) 9 min

Figure 3.7 shows streamlines at different times. Melting of Solid Gallium takes place due to the

heated wall and a solid-liquid interface moving in the right hand side of the cavity and therefore,

density of the liquid Gallium changes with temperature in the area adjacent to the heated wall. The

liquid Gallium rises up having less density and heavier liquid stays at the bottom. As a result, a

convection current due to density difference in liquid Gallium is set up inside the cavity, which iscalled as natural circulation, and this enhances the melting.

Figure 3.8 shows temperature contours at different times. Initially, the contours are straight which

indicates that heat transfer occurs mainly due to conduction. But as the time progresses, convection

phenomena becomes dominant and a slight curvature in the contour plot is observed.


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Figure 3.9 shows interface comparison with Brent et al. [13] at different times. Front position

comparison obtained is satisfactory. To plot the interface position, fl=0.5 contour is used. At any

particular time, the interface divides the cavity area into two distinct phases.

With the passage of time, front moves rapidly in the top but the movement is very slow in the

bottom. This signifies that the convection is more prevalent in the upper portion. Hot liquid

impinges on the solid Gallium in the top of the cavity and therefore melting rate is more in the top

as compared to

(a) (b)

Figure 3.8:-Temperature Contours for Gallium Melting (a) 6 min, (b) 9 min

Figure 3.9: Interface position at different times


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the bottom portion.

However, melting of Gallium is a controversial problem in the literature [26]. There is a

controversy between multi-cellular and mono-cellular liquid flow of Gallium, first pointed out by

Dantzig [27]. In the present study, mono-cellular liquid flow is assumed. A great effort has been

made by Hannoun et al. [26] to solve the controversy.

However, the interface position does not match well with the literature. It overestimates the result

given in Brent et al, the reason could be time step used and discretization techniques. Due to

controversial effect discussed in the previous paragraph, may also play an important role in the

estimation of the interface position at different times.

3.2.2 Gallium Melting Without Convection

Gallium melting as already stated, being a benchmark problem for isothermal phase change

problems, in which now the convection effect earlier considered and was neglected and the suitable

data value are obtained and studied.Again the same problem from Brent et al. [13] is dealt with convection effect neglected. For the

convenience to reader again the problem is defined in the same manner as it was done in the

previous problem.

Figure 3.10:- Melting of Gallium Problem

Similar post processing is done as in Gallium melting with convection problem, with the aim tocorrelate both the problems deduce the dominating factor prevalent in the problems.


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(a) (b)

Figure 3.11:Interface for Gallium melting (a) 6 min, (b) 9 min

Figure 3.11 shows interface at different times. Melting of Solid Gallium takes place due to the

heated wall and a solid-liquid interface moves in the right hand side of the cavity. Density of theliquid Gallium changes with temperature in the area adjacent to the heated wall. Since the

convection effect is neglected no circulation phenomenon is developed.

Figure 3.12 shows temperature contours at different times. All through the simulation period,

contours are straight which indicates that heat transfer occurs mainly due to conduction. Since,

only convection predominates only straight contours are observed, with negligible streamlines.

(a) (b)

Figure 3.12: Temperature Contours for Gallium Melting (a) 6 min, (b) 9 min


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3.3 Mushy region problem

Till now, we have dealt with isothermal phase change problems with and without natural

convection. In case of isothermal phase change, phase change occurs at distinct temperature. But,

in case of mushy region, phase change occurs over a temperature range. Therefore, latent heat isdependent upon the temperature. Several relationships between latent heat and temperature can be

possible. A linear relationship is mostly used in literature due to its simplicity. In the present case,

both linear and non linear relationships are taken. In isothermal phase change, we have neglected

the convective term of the latent heat, but it is included in mushy region [12].

Both two and three dimensional problem are presented here.

The linear relationship can be obtained as follows:

= + (3.1)Where a and c are constants. Now, the value of the constants can be determined by applying

suitable conditions. At T=Ts, eL=0 and at T=TL, eL=. By applying these condition the following

relationship is obtained

= ( ( ) ; (3.2)

A more general relation is

= ( ) ; (3.3)

Where n is the index value. Generally, 2 n 5 is accepted. When n=1, the relationship becomes

linear. Otherwise, it is non-linear.

3.3.1 Two-dimensional problem

A two dimensional mushy region phase change problem is taken from Vollar and Prakash [12].However, it has been reported in [24] that for high Prandtl number (Pr=1000) liquid, the

requirement of computational time is more. Therefore, to reduce the computation time, low Prandtl

number (Pr= 10) is taken by Debraj Das [24].

We have also taken Pr=10 in the simulation. The computational domain is shown in Fig.

3.13. A 40 X 40 uniform hexahedral grid is taken in 1 x 1 domain. Initially, cavity is filled with

liquid having initial temperature 0.5C. Suddenly, left wall temperature is reduced to -0.5C and

right wall is kept as the initial temperature 0.5C. Suddenly, left wall temperature is reduced to -


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0.5Cand right wall is kept as the initial temperature. Other two boundaries are kept insulated. The

problem is solved with different values of .The value of the liquid properties shown in Table 3.4

are calculated keeping the non- dimensional number same (Ra=104, Pr=10, St=5.0)[24]. In the

present computation, the value of constant A and b is taken as 103

and 0.01. Figure 3.14 shows

Figure 3.13:-Mushy region two dimensional problem

Table 3.4:-Physical Properties taken for two dimensional mushy region problem

k Cp Tf

(W/m-K) (J/kg-K) (Kgm-

) (J/kg) (kg/m-s) (1/K) (K)

Solid 0.001 1.0 1.0 0.0 - - -

Liquid 0.001 1.0 1.0 5.0 0.01 0.01 0.0

Vector plot along with the position of mushy region at different times. For simulation validation

the value of is taken as 0.1. The mushy zone is the region bounded by the solidus and liquidus

lines. Therefore, region bounded byTL= andTS= -is the mushy zone as = (TL- TS)/2. It is

seen from the vector plot that fluid velocity is not zero inside the mushy zone, so the mushy zone

contributes some convection which is expected.

As time progresses mushy zone moves towards the right wall and the solidification rate is

increased. However, in solidification there can exist a steady state [24]. In this problem, it is seen

that mushy zone does not move very much with time after t = 1000 sec. So, the problem reachessteady state. In the present section, unsteady results


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(a) (b) (c)

Figure 3.14:- Vector plot and mushy region for =0.1 (a)t=100 sec, (b)t=600 sec, (c)t=1000 sec

are presented only.

Comparison of uvelocity at different horizontal sections at t = 500 sec is shown in Figure 3.14(a).Figure 3.14(b) shows the comparison of solidus and liquidus line at t =1000sec. The comparison is

good. The velocity variations are sinusoidal in nature. Small Variation in results is found due to

difference in grid.

Figure 3.16 shows the temperature contours at different times. Near to the left wall contours are

straight which indicates that the heat transfer mainly occurs due to conduction only.

Figure 3.15(b) shows the effect of half mushy range () on the width of the mushy region at t=

1000 sec.

(a) (b)

Figure 3.15:- Comparision of (a) uvelocity at t= 500 sec, (b) solidus and liquidus line att=1000


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(a) (b)

Figure 3.16:-Temperature contours for two dimensional mushy region problem (a) t=600 sec, (b)

t=1000 sec

The problem has been solved by assuming a linear relationship between latent enthalpy andtemperature.


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3.4 Practical Application (Continuous casting process)

Continuous casting process is now-days, a backbone manufacturing process for almost all the

manufacturing firms, especially the steel plants. With advancement in technology, we have seen

the use of enhanced and improved manufacturing methods, to eliminate earlier defects like

porosity, blow holes, air inclusion, slag entrapment and many more. Here, therefore we have

implemented Ansys Fluent to simulate the real life problem virtually, to effectively study the

outcomes in order to draw conclusions and eradicate the problems faced in manufacturing

industries.

Here, we are solving a benchmark problem on Solidification and melting commonly known as

Solidification in Czochralski Model, which takes place in Continuous Casting Process and is

directly taken from Ansys Fluent tutorial, defined as a solidification problem which involve fluid

flow and heat transfer problem. A 2D axis symmetric bowl has been considered as geometry as

shown in the Figure 3.17 which contain the liquid, the boundary conditions have been mentioned

on the figure itself. The bottom and the sides of the bowl are heated above the liquidus

temperature, as it is the free surface of the liquid. The liquid is solidified by heat loss from the

crystal and the solid is pulled out of the domain at a rate of 0.001 m/s and a temperature of 500 K.

A steady injection of liquid is maintained at the bottom of the bowl with a velocity of 1.01 x 10-3

m/sand a temperature of 1300 K. Material properties. Initially, steady solution is computed and

then the fluid flow is enabled to investigate the effect of natural and Marangoni Convection in a

transient fashion.

As in continuous casting the material is pulled out continuously so, pull velocity is enabled.

A plot of temperature contours is shown in Figure 3.18, in accordance with the literature present

in Ansys Fluent Tutorial [23].

Contours of temperature (Mushy zone) is shown in Figure 3.19-: in steady state solution,


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Figure 3.17: Solidification in Czochralski Model

Case 1: Steady State Solution

In steady state solution we would specify the type of discretization form, and would restrict the

calculation of flow and swirl velocity equation on temporary basis, in order to calculate

conduction only.

Figure 3.18:-Shows the temperature contours for steady conduction solution.


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Figure 3.19:- Shows Contours of Static Temperature (Mushy Zone) in steady State.

Case 2: Transient State Solution

The previous steady state solution was taken as the initial condition for transient state solution.

In this flow and swirl velocity equation is being calculated.

Now, for transient state solution numerous plots have been plotted in various parameters to

validate the problem with the thesis [23].

Figure 3.20:- Shows the Static temperature contour in transient state


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Figure 3.21:- Contours for Stream function at t = 0.2 sec

In the Figure 3.21 , the contour obtained for Stream function is due to Natural convection andMarangoni convection on the free surface.

Figure 3.22-:Contours for liquid fraction

Figure 3.22, here the current position of melt front is being displayed here. The Mushy zone

divides the liquid and solid regions roughly into half.

Further simulation is done for 48 more number of time steps in all total of 50 time steps. And the

plots obtained us somewhat like this. In Continuous Casting, it is very important to know when to

pull the material out. If it is pulled earlier it wont get solidified (still in mushy zone), and if

pulled late it will be solidified and wont be pulled out in a desired shape. By proper study of the

solidus and liquidius temperature contour, the optimal pull rate can be easily calculated.


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Figure 3.23:- Contours of temperature t = 5 sec

The Figure 3.23, shows that the temperature contour is fairly uniform along the melt zone and

solid material, and due to recirculating liquid the distortion obtained in the temperature is clearly

visible.

After 5sec the flow is mainly dominated by Natural convection and Marangoni stress as seen

from Figure 3.24.

Figure 3.24: Stream function Contours at t=5 sec


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Figure 3.25:- Contours of liquid fraction at t= 5sec


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4. Conclusions and scope for future work

4.1 Conclusion

The following study entrails the simulation done through Ansys Fluent, on few

benchmark solidification and melting problems which include 2D and 3D isothermal problem,

Gallium melting with and without convection, a Mushy region and problem and in the last a

application problem on Continuous casting problem named as Solidification in Czochralski

Model, and has been validated with the numerical and analytical data present in the literature.

In the first two problems for isothermal case (diffusion problem) has been solved, followed by

mushy region problem.

In Gallium melting with and without convection has been tried to solve to show that convection

plays a vital role in enhancing the rate of melting. Many Plots on temperature, streamlines and

interface has been plot to show the variation obtained. Natural Convection pattern has been

shown using the streamline plot.

In the last problem, on continuous casting, both steady state and transient state solution has been

shown.

4.2 Scope for Future Work

We have neglected Density Change in the present thesis. With rapid void and Shrinkage

formation because of Density change the flow field changes to shrinkage-induced flow. In thepresent study species equation has not been solved. So, one could add with Navier-Strokes and

Energy equation for determining the composition in each phase in the multi-component alloy

solidification process. Only thermal buoyancy has been considered. However, due to non-

uniform cooling solutal buoyancy effect may arise. The flow has been considered as laminar.

Models related to turbulent flow can also be added to phase change problems. The Ansys

Simulation results has been verified with the literature data.


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5. References

[1] H. Hu and S. A. Argyropoulos, Mathematical modelling of solidification and melting,

Modelling Simul. Mater. Sc. Engineering, vol. 4, pp. 371396, 1996.[2] H. T. Hashemi and C. M. Sliepcevich, A numerical method for solving two-dimensional

problems of heat conduction with change of phase, Chemical Engineering Progress Symposium,

vol. 63, no. 79, pp. 3441, 1967.

[3] D. Poirier and M. Salcudean, On numerical methods used in mathematical modeling of

phasechange in liquid metals, ASME Journal Heat Transfer, vol. 110, pp. 562570, 1988.

[4] B. Basu and A. W. Date, Numerical modelling of melting and solidification problems-a

review, Sadhana, Part 3, vol. 13, pp. 169213, 1988.

[5] A. Lazaridis, A nemerical solution of the multidimensional solidification (or melting)

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[6] V. R. Voller and M. Cross, Accurate solutions of moving boundary problems using enthalpy

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[9] C. Gau and R. Viskanta, Melting and solidification of a pure metal on a vertical wall,

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[10] K. Morgan, A numerical analysis of freezing and melting with convection, Computer

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phase change, Int. J. Num. Methods Engg, vol. 24, pp. 271284, 1987.

[12] V. R. Voller and C. Prakash, A fixed grid numerical modelling methodology for

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[13] A. D. Brent, V. R. Voller, and K. J. Reid, Enthalpy-porosity technique for modeling

convection-diffusion phase change: Application to the melting of a pure metal, Numerical

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[14] F. Wolff and R. Viskanta, Solidification of pure metal at a vertical wall in the presence of

liquid superheat, Int. J. Heat Mass Transfer, vol. 31, pp. 17351744, 1988.[15] M. A. Rady and A. K. Mohanty, Natural convection during melting and solidification of

pure metals in a cavity, Numerical Heat Transfer Part A, vol. 29, pp. 4963, 1996.

[16] F. Stella and M. Giangi, Melting of a pure metal on a vertical wall: Numerical simulation,

Numerical Heat Transfer Part A, vol. 38, pp. 193208, 2000.

[17] A. M. Rady, V. V. Satyamurty, and A. K. Mohanty, Effects of liquid superheat du ring

solidification of pure metals in a square cavity, Heat and Mass Transfer, vol. 32, pp. 499509,


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1997.

[18] B. Ghasemi and M. Molki, Melting of unfixed solids in square cavities, International

Journal of Heat and Fluid Flow, vol. 20, pp. 446452, 1999.

[19] Z. Gong and A. S. Mujumdar, Flow and heat transfer in convection dominated melting in a

rectangular cavity heated from below, Numerical Heat Transfer Part B, vol. 23, pp. 2141,

1993.

[20] O. Bertrand, B. Binet, H. Combeau, S. Coutturier, Y. Delannoy, D. Gobin, M. Lacroix, P. L.

Quere, M. Medale, J. Mencinger, H. Sadat, and G. Vieira, Melting driven by natural convection

a comparison exercise:first results, Int. J. Them. Sci, vol. 38, pp. 526, 1999.

[21] K. Hwang, Effects of density change and natural convection on the solidification process of

a pure metal, J. Materials Processing Technology, vol. 71, pp. 466476, 1997.

[22] A. Dalal, V. Eswaran, and G. Biswas, A finite-volume method for navier-stokes equation

on unstructured meshes, Numerical Heat Transfer, Part B, vol. 52, pp. 238249, 2008.

[23] Ansys Tutorial 2008 Full Guide, Melting and Solidification Problem, Chapter 24,

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