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 http://digital.auraria.edu/AA00005688/00001
Material Information
 Title:
 Influence of magnetic field on direct contact melting undergoing rotation
 Creator:
 Malepati, Srikesh
 Place of Publication:
 Denver, Colo.
 Publisher:
 University of Colorado Denver
 Publication Date:
 2016
Thesis/Dissertation Information
 Degree:
 Master's ( Master of Science)
 Degree Grantor:
 University of Colorado Denver
 Degree Divisions:
 Department of Mechanical Engineering, CU Denver
 Degree Disciplines:
 Mechanical engineering
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 Source Institution:
 University of Colorado Denver Collections
 Holding Location:
 Auraria Library
 Rights Management:
 Copyright Srikesh Malepati. Permission granted to University of Colorado Denver to digitize and display this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

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INFLUENCE OF MAGNETIC FIELD ON DIRECT CONTACT MELTING
UNDERGOING ROTATION by
SRIKESH MALEPATI B.Tech, Gitam University, 2014
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering Program
2016
This thesis for the Master of Science degree by Srikesh Malepati has been approved for the Mechanical Engineering Program by
Kannan Premnath, Chair Samuel Welch Dana Carpenter
December 17, 2016
n
Malepati, Srikesh (M.S., Mechanical Engineering)
Influence of Magnetic Field on Direct Contact Melting Undergoing Rotation Thesis directed by Assistant Professor Kannan Premnath
ABSTRACT
Contact melting heat transfer is a major industrial process which has diverse applications in various engineering fields. It is a physical process achieved by forced contact between a heating source and a phase change material undergoing melting. Contact melting processes are classified based on many factors like the relative motion between heating material and melting substance, location of the phase change interface and motion of heating substance etc. Previously there have been numerous studies on various factors which affect the melting rate in contact melting process. Magnetohydrodynamics has gained importance in recent years in the engineering field. The effect of magnetic forces on contact melting is an interesting area of research. This thesis deals with determining the effects of magnetic fields on close contact melting processes subjected to rotation. Computational domain in this work is described as a melting substance placed on rotating heater and is subjected to external magnetic fields. Governing equations are developed for this problem by referring to previous literature, and additional forces induced because of the magnetic fields are indicated as additional terms in the governing equations. Various characteristic nondimensional variables are defined and all governing equations involving three dimensional mass, momentum and energy equations in cylindrical coordinate system coupled with an interfacial transport equation are transformed into such nondimensional variables leading to a set of similarity equations. Finite difference methods are employed in solving the transformed governing equations. Externally specified variables like rate of circular rotation and strength of magnetic field have also been nondimentionalized and are referred to as external force parameter a and magnetic parameter M. Nu
merical results are then obtained from the solution of the similarity equations and the effect of various governing nondimensional variables on the contact melting and heat processes undergoing rotation in the presence of a magnetic held are studied systematically and interpreted. In particular our results reveal that melting rate and heat transfer rate increase with increase in a, while such rates decrease with increase in M.
The form and content of this abstract are approved. I recommend its publication.
Approved: Kannan Premnath
IV
This thesis is dedicated to ALMIGHTY (SHIVA), to MY PROFESSOR and to MY FAMILY.
v
ACKNOWLEDGMENTS
Firstly I would like to thank my professor and advisor Dr Kannan Premnath for defining my thesis problem statement and guiding me in solving it. Without his constant support this would not be possible. I would like to also thank Dr Samuel Welch and Dr Dana Carpenter for agreeing to be a part of my thesis committee.
Very special and sincere thanks to Farzaneh Flajabdollahi for helping me understand, develop and modify the MATLAB code to solve my thesis problem statement.
I wish to thank Bhanu Babaiahgari for his assistance with LaTex and other computing issues.
I would also like to appreciate Kenneth Sisco and his team from engineering computer lab for providing me all kinds of computing services.
Special thanks to my FAMILY and FRIENDS for supporting me through out my masters program.
vi
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION .................................................................. 1
1.1 Introduction............................................................. 1
1.2 Literature Review on Close contact melting............................... 1
1.3 Literature Review on Close Contact Melting with Rotating Disk . 2
1.4 Literature Review on Magnetohydrodynamics................................ 2
1.5 Present Work ............................................................ 3
II. MATHEMATICAL MODELLING........................................................ 5
2.1 Introduction............................................................. 5
2.2 Computational Domain Description......................................... 5
2.3 Magnetic Effects......................................................... 7
2.4 Governing Equations ..................................................... 8
III. NONDIMENSIONAL PARAMETERS ................................................. 10
3.1 Introduction............................................................ 10
3.2 NonDimensional Variable................................................ 10
3.3 Transformed Governing Equations......................................... 11
3.4 Finite Difference Method................................................ 15
IV. RESULTS AND DISCUSSION ..................................................... 18
4.1 Introduction............................................................ 18
4.2 Velocity and Temperature profiles within Melt Layer..................... 19
4.2.1 Radial velocity component.......................................... 19
4.2.1.1 Effect of external force on radial velocity of the melt layer 20
4.2.1.2 Effect of magnetic field on radial velocity of the melt layer 22
4.2.2 Axial velocity component........................................... 24
4.2.2.1 Effect of external force on axial velocity of the melt layer 25
4.2.2.2 Effect of magnetic field on axial velocity of the melt layer 27
vii
4.2.3 Angular velocity component....................................... 29
4.2.3.1 Effect of external force on angular velocity of the melt
layer..................................................... 30
4.2.3.2 Effect of magnetic fields on angular velocity of the melt
layer..................................................... 32
4.2.4 Temperature distribution......................................... 34
4.2.4.1 Effect of external force on temperature distribution of
melt layer................................................ 35
4.2.4.2 Effect of magnetic fields on temperature distribution of
the melt layer............................................ 37
4.3 Melt Film Thickness.................................................. 39
4.3.1 Effect of external force on melt the layer thickness ............ 40
4.3.2 Effect of the the magnetic field on the melt layer thickness . 43
4.4 Melting Rate........................................................... 46
4.4.1 Effect of external force on melting rate......................... 46
4.4.2 Effect of magnetic field on melting rate......................... 49
4.5 Heat Transfer Rate .................................................. 52
4.5.1 Effect of external force on heat transfer rate at the heater wall 53
4.5.2 Effect of external force on heat transfer rate at the phase change
location........................................................ 56
4.5.3 Effect of magnetic field on heat transfer rate at the heater wall 58
4.5.4 Effect of magnetic fields on heat transfer rates at the phase
change location................................................. 61
4.6 Tangential Shear Stress and Radial Shear Stress........................ 63
4.6.1 Tangential shear stress.......................................... 64
4.6.1.1 Effect of external force on tangential shear stress at the
heater wall............................................... 64
viii
4.6.1.2 Effect of external force on tangential shear stress at the
phase change location................................ 66
4.6.1.3 Effect of magnetic field on tangential shear stress at the
heater wall.......................................... 69
4.6.1.4 Effect of magnetic field on tangential shear stress at the
phase change location................................ 71
4.6.2 Radial shear stress........................................ 74
4.6.2.1 Effect of external force on radial shear stress at the
heater wall.......................................... 74
4.6.2.2 Effect of external force on radial shear stress at the phase change location.............................................. 76
4.6.2.3 Effect of magnetic field on the radial shear stress at the heater wall.................................................. 79
4.6.2.4 Effect of magnetic field on radial shear stress at the
phase change location................................ 81
V. SUMMARY AND CONCLUSION................................................ 85
REFERENCES................................................................ 88
IX
FIGURES
2.1 Computational domain for the contact melting problem undergoing rotation in the presence of magnetic held.................................. 7
4.1 Variation of radial velocity of the liquid melt layer as a function of exter
nal force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.......... 21
4.2 Variation of radial velocity of the liquid melt layer as a function of mag
netic held parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.... 24
4.3 Variation of axial velocity of the liquid melt layer as a function of exter
nal force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.......... 26
4.4 Variation of axial velocity of the liquid melt layer as a function of magnetic
held parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.......... 29
4.5 Variation of angular velocity of the liquid melt layer as a function of ex
ternal force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.......... 31
4.6 Variation of angular velocity of the liquid melt layer as a function of mag
netic held parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.... 34
4.7 Variation of the temperature of the liquid melt layer as a function of ex
ternal force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.......... 36
4.8 Variation of the temperature of the liquid melt layer as a function of magnetic held parameter M for different Prandtl numbers Ste when external
force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100...... 39
x
4.9 Variation of the thickness of the liquid melt layer as a function of exter
nal force parameter a for different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100............. 41
4.10 Variation of thickness of the liquid melt layer as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter
M = 5 and Pr=10........................................................ 42
4.11 Variation of thickness of the liquid melt layer as a function of magnetic
parameter M for different Stefan numbers Ste when external force parameter
4.12 Variation of the melting rate as a function of external force parameter a for
different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100....................................... 48
4.13 Variation of the melting rate as a function of external force parameter a
for different Stefan numbers Ste when magnetic parameter M = 5 and Pr=10............................................................... 49
4.14 Variation of the melting rate as a function of magnetic parameter M
for different Stefan numbers Ste when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.............................. 51
4.15 Variation of the heat transfer rate evaluated at heater wall as a function of
external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100............. 54
4.16 Variation of heat transfer rates evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste
when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100 57
4.17 Variation of heat transfer rates evaluated at heater wall as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter a = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100 .... 60
xi
4.18 Variation of heat transfer rates evaluated at phase change location as a
function of magnetic parameter M for different Stefan numbers Ste when external force parameter <7 = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100 62
4.19 Variation of tangential shear stress evaluated at wall as a function of
external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100............ 65
4.20 Variation of tangential shear stress evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50,
(d)Pr=100)............................................................ 68
4.21 Variation of tangential shear stress evaluated at wall as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter Ste = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100) . . 70
4.22 Variation of tangential shear stress evaluated at phase change location
as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter Ste = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50,
(d)Pr=100)........................................................ 73
4.23 Variation of radial shear stress evaluated at wall as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100)............... 75
4.24 Variation of radial shear stress evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100 78
4.25 Variation of radial shear stress evaluated at heater wall as a function of
magnetic fields M for different Stefan numbers Ste when external force parameter a = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.................... 80
Xll
4.26 Variation of radial shear stress evaluated at phase change location as a function of magnetic fields M for different Stefan numbers Ste when external force parameter a = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100 . 83
xm
CHAPTER I
INTRODUCTION
1.1 Introduction
Contact melting is a physical thermal transport process and accompanied by a fluid motion in a thin region, when there is a forced contact between a solid phase change material (PCM) and a heater. Typically, the heater is maintained at high temperatures. The heater can be subjected to constant heat flux or it is maintained at a constant temperature. The solid PCM may vary based on the type of melting process, for example ice block for melting of ice related process and solid block made of metal in manufacturing industries. Application of contact melting can be fairly found in all engineering industries like manufacturing, aerospace, ship building etc.
1.2 Literature Review on Close contact melting
Close contact melting is always characterized by a liquid him formation between the rubbing solid surfaces. Development of liquid him is achieved by its own viscous friction from its previous layer of melting solid [1]. The mode of heat transfer between the liquid him formed is primarily related to heat conduction. Convection heat transfer are dominated in liquid hlms which have larger magnitude of thickness. In most of the cases of close contact melting the liquid him thickness which separates heater and PCM solid are small, since the melt removal is achieved with multiple processes [2], Contact melting process can be widely generalized as close contact melting and hxed contact melting. There is a relative motion between the source of heat and melting solids in close contact melting. Either the heat source can be motion or the melting substance is in motion [3]. Close contact melting offers mainly two advantages over hxed contact melting, i.e the liquid melt formation is prevented because of the relative motion and the melt is at a lower average temperature in the system. Internal close contact melting is dehned when the Phase Change Material melts inside the heater
source enclosure, whereas the external close contact melting is dehned when the PCM
1
melts around an embedded heat source body [4]. Our work mainly focuses on external close contact melting case. Close contact melting applications are spread across wide range of engineering applications. Some of the typical applications involve selfburial waste disposals of radioactive elements, reactor core melt down in nuclear industries [5], applications in geophysics, spacecraft and material processing industries [2], Contact melting processes inside closed enclosures are important liquidsolid phase change processes, which are used in technology applications where large amounts of energy transfer take place. For example, air conditioning pumps, latent heat thermal energy storage systems, solar water heating conductors and heat pumps [6].
1.3 Literature Review on Close Contact Melting with Rotating Disk
Melt removal formed in between the heater plate and phase change material is important to enhance the heat transfer rate between the solid bodies. The thinner the melt layer thickness is, the larger is the heat transfer rate through heat conduction [7]. There are various methods of melt removal methods, like the use of magnetic fields and gravitational forces [8]. Melt removal process using the weight of melting solid was also studied previously [9]. In all the cases of contact melting there is a basic phase change of melting substance. Rotating heater is employed in reduced gravity experiments, where the melt removal is achieved with the centrifugal forces generated by rotating heater [10]. Material processing applications carried out in microgravity experiments are also some of the useful applications of rotating disk contact melting. Many authors have analyzed and studied the governing equations of melting solids under rotation. Exact numerical and analytical solution were also developed for determining the melting rate [10].
1.4 Literature Review on Magnetohydrodynamics
Magnetohydrodynamics (MHD) is the study of the interaction of magnetic effects on the hydrodynamic properties of moving fluids. There has been extensive study
2
made on magnetic effects on flow parameters and heat transfer rates. Most studies in MHD have been carried out to determine the hydrodynamic stability during certain boundary conditions [11, 12]. References [13, 14] have studied MHD effects on heat transfer rates in rotating disks and natural convection studies respectively. Primary applications of MHD flows on rotating disks include high speed turbine maintenance, rotary part systems and cooling of nuclear reactors. Its theoretical applications include space physics, geophysics and astronomy. Initial ideas of flow around the rotating disks in the absence of magnetic field effects were pioneered by Von karmann in (1921) [15]. He developed a model with governing equations which capture all of the essential physics involved in flow undergoing rotation. His approach towards this problem with similarity solutions resulted in terms of nondimensional variables, which simplified the nonlinear governing equations to ordinary differential equations. Such a approach can be extended to MHD flow over rotating disks [13]. In general it is found that in various thermal transport processes, a decrease in the fluid velocities and heat transfer rates occur because of the presence of magnetic field [16].
1.5 Present Work
Combined effects of magnetic fields and rotation have been a potential area of research. As per the knowledge of the author there is no prior study in the literature on contact melting processes when subjected to both rotation, magnetic fields and external pressure by melting solid. Through insights into such effects achieved by a detailed modeling of the complex nonlinear processes involved, one may then devise control techniques to control the qualities of contact melting procedure. This research mainly addresses the influence of magnetic field on direct contact melting process undergoing rotation. In this regard, a mathematical model involving three dimensional mass, momentum and energy equations in cylindrical coordinates coupled with and interfacial phase change condition is developed for this rotating system. Magnetic
field effects are accounted for by means of the additional body force terms. Using ap
3
propriate nondimensional similarity transformation, the governing partial differential equations are then transformed into a set of similarity ordinary differential equations. A nonlinear finite difference solver for the similarity equations representing flow and thermal transport in rotating system and accompanied by phase change and magnetic held effects are developed. All the velocity components of the melt layer, heat transfer rate between the heating surface, melt and melting solid, rate of melting and thickness of the melt layer are studied for varying values of magnetic fields and rotational speeds in terms of characteristic nondimensional parameters. Some important physical parameters of engineering interests like tangential shear stress and surface shear stress in the radial direction of the rotating heater are also studied. This complete dissertation is divided into five chapters. This chapter primarily deals with literature review on the problem. Chapter 2 details about the mathematical approach taken towards studying the contact melting due to combined effects of rotation and magnetic fields. It presents the governing equations in cylindrical coordinates. All the nondimensional parameters are defined in chapter 3 which also discusses the similarity transformations and numerical solution procedure on finite difference method. Chapter 4 presents all key results and discussions and chapter 5 provides a summary and conclusions arising from this thesis work.
4
CHAPTER II
MATHEMATICAL MODELLING
2.1 Introduction
This chapter deals with the description of the analytical approach used to model the contact melting problem subjected to both magnetic fields and rotation. The physical problem consists of a solid melting block, considered as the phase change material (PCM) placed on rotating disk heater. The whole setup is subjected to an external magnetic held. For ease of analysis the phase change material is maintained at its melting temperature and the rotating heater is maintained at a temperature higher than the melting temperature of the phase change material. As the contact melting process continues, the phase change material undergoes melting and it is separated from heater with a thin layer of melt being formed in between them. Melt removal is achieved with a centrifugal force generated from the rotation of heater. Liquid melt which is flowing outwards, has various velocity components in different directions of the coordinate system. Effects of the rate of rotation of the heater along with the applied external magnetic held is also studied on these velocity components as well as those on the melting rate and thickness of melt layer formation and heat transfer rates. Consistent and simplified forms of the Maxwells equations are then used to obtain the induced magnetic fields, which results in the Lorentz force influencing the dynamics of the thin liquid him generated by the contact melting process.
2.2 Computational Domain Description
A schematic presentation of the computational domain considered is described
in the Fig. 2.1. A melting solid, phase change material is described in the form of
parallelepiped cylinder, and its dimensions are dehned as L x r0, where L is the initial
height and r0 is radius of melting solid. It is maintained at its melting temperature
Tm. This is placed on a cylindrical heater which is maintained at a temperature Tw
and rotated with an angular rotation rate oj. Thickness of the melt layer in the contact
5
melting process is defined as 8 and the melt layer initially covers a radius ro and then flows outwards. A vertical downward force which is weight of the solid is acting on the melting solid which also helps in the liquid melt removal process in addition to centrifugal force generated due to rotation. The pressure developed because of the weight of the melting solid is considered as p0 and generates the squeezing effect. An external magnetic field is applied to this setup, which acts perpendicular to the direction of liquid melt removal as shown in Fig 2.1. The strength of this magnetic field is denoted as B0. This external magnetic field generates various force components in different directions which will be discussed in later part of this chapter.
We consider the following assumptions to proceed further for developing the governing equations
Flow of the liquid melt which is directed radially outward is laminar
By looking at the computational domain we can conclude azimuthal symmetry. This indicates we can neglect the ^ terms in governing equations.
Heater is maintained at constant temperature described earlier.
Melting process is assumed to be quasisteady.
Melt liquid layer generated is considered to be incompressible, viscous and Newtonian fluid.
6
Magnetic field
4
4
Figure 2.1: Computational domain for the contact melting problem undergoing rotation in the presence of magnetic held
2.3 Magnetic Effects
All the effects related to the magnetohydrodynamics are understood with three
important laws namely Faradays law of induction, Amperes law and Lorentzs force.
The coupling between the contact melting process and magnetic held can be studied
by the combined effects of huid mechanics and electromagnetic effects. They also
involve an energy transport phenomenon at the phase change location due to latent
heat effects and also in the liquid melt via conduction and convection. Relative
movement from the liquid melt and applied external magnetic held induces e.m.f
(electromotive force) due to the Faradays law. Underlying concepts induced by the
external magnetic held can be captured by the addition of a force term into the
7
governing equations as mentioned earlier. This takes the form F = J x B where J is the current density and B represents the magnetic induction vector. The induced current density J which can be defined via Ampere law of magnetohydrodynamics as J = cre(E + V X B) Here E is the electric held strength, V X B is the induced current and ae is the electrical conductivity. Previous studies and literatures have shown that we can neglect the creE term which then simplifies the magnetic force term referred to the Lorentz force as F = cre (V X B) This additional force has a radial component and a circumferential component, which mainly affect the how parameters of liquid melt [13]. Thus Fr. = creVr.B02 is the radial component and while F^ = creV^_B02 represents the circumferential component of Lorentz force. Both of these terms are included in the governing equations to include the physics generated by external magnetic helds in the next section.
2.4 Governing Equations
Basic fundamental equations which govern the problem statement are the mass continuity equations, momentum continuity equations and energy equation. Physics at the interface due to the phase change arising via latent heat effects can be captured with Stefan condition.
To facilitate modelling and analysis, we represent the governing equations in the threedimensional cylindrical coordinates, in which the velocity held is written as V = (Vr, V'f,, Vz). First we consider the mass continuity equation represented by [10]
!Â£*> +(2.i)
r or dz
The components of the Navier stokes equations including the Lorentz force effects can then be written as
dvr
dr
dvr
' dz
J4>
1 dp p dr
v
d2Vr 1 dVr d2Vr
dr2 r dr
dz2
(JeVrBQ
P
(2.2)
dvp
dr
dv,
'(f) VrV(f)
dz
d2vp 1 dvp d2v(f)
dr2 r dr dz2
f2
(TeVpBl
(2.3)
(2.4)
dvz
dr
dvz
' dz
1 dp
p dz
v
' d2vz dr2
1 dvz r dr
d2vz dz2
Here, Eq. (2.2) is the momentum equation in radial direction, Eq. (2.3) in the angular direction and Eq. (2.4) in the axial direction. vr is the velocity defined in radial direction, v^ represents the velocity defined in circumferential direction and vz is the velocity defined in axial direction, p = p(r, z) is the local pressure in the liquid melt layer, whose cumulative effect balances the load acting on the melting solidP0The energy balance equation is then represented as
dT dT
Vrzr +vzzpdr dz
a
d2T 1 dT d2T
dr2 r dr dz2
(2.5)
Based on the computational domain, boundary conditions for this problem are defined at two different locations. One at the heater wall (z = 0), and another at the phase change location i.e at the interface (z = 8). Boundary conditions which are applied to the to the computational domain are as follows.
vr = 0 = ru vz = 0 T = Tw at z = 0
(2.6)
vr = 0 = 0 T = Tm at z = 5 (2.7)
dT
~k\z=s = phsfVz (z = 8) (2.8)
Here Eq. (2.8) is the energy balance at the interface location, i.e the Stefan
condition. In the above equations, the fluid properties p, u, a, k and hsf represent the density, kinematic viscosity, thermal diffusivity, thermal conductivity and latent heat of melting respectively.
9
CHAPTER III
NONDIMENSIONAL PARAMETERS
3.1 Introduction
Governing equations for contact melting problem undergoing rotation and subjected to magnetic fields are partial differential equations (PDEs). Solving those partial differential equations requires specialized numerical methods and considerable computational resources. Converting them into ordinary differential equations (ODEs) would simplify the solution procedure and significantly reduce requirements of computational resources. Nondimensional similarity variables are used in converting the partial differential equations to ordinary differential equations. This primary approach of implementing the nondimensional parameters to the rotating disk problem was first developed by Von Karman in his classical viscous pump flow problem analysis. The problem is concerned about the flow of semiinfinite fluid on very large flat disk which is being rotated at a particular circular frequency. Our problem statement is similar to Von Karman viscous pump. But our thesis work deals with finite liquid him thickness which is different from semiinfinite setup in viscous pump. We adopt similar approach to convert our partial differential equations to ordinary differential equations with changes in boundary conditions; also for a finite domain in the radial direction by taking into consideration the phase change and magnetic held effects.
3.2 NonDimensional Variable
The list of primary nondimensional variables used in our analysis to simply and solve our governing equations of the physical model are as follows
(3.1)
F(r/)
(3.2)
10
G(,,) = 'i
ruj
Vz
H(v)
P(v)
0(v)
(uuj)1/2
[P{r, z) p(r, Â£)] pvoj T T
JW L
T T
J in J m.
(3.3)
(3.4)
(3.5)
(3.6)
3.3 Transformed Governing Equations
As discussed earlier after applying these transformation variables to the governing equations mentioned in the previous section, we can transform them to ordinary differential equations ODEs. We first apply the nondimensional variables to the mass continuity Eq. (2.1) giving
1 d{rFruj) d
r dr dz
which after differentiating yields
2 ujF + ujH' {viS)1!2 (oj j v)1^ = 0 After simplifying this we get following similarity equation
(3.7)
(3.8)
(3.9)
We apply similar kind of approach when we aim to replace the momentum governing equation with their corresponding nondimensional variables. We now mainly focus our attention to governing Eq. (2.2). As part of this derivation, we will introduce a nondimensional parameter a characterising external loading and rotation effect and nondimensional magnetic hied parameter M.
To begin derivation that introduces the external force parameter a, we first neglect magnetic term and redefine pressure term in the problem statement. From the
previous literature it has been concluded that the pressure is independent of radius
11
H' = 2 F
[10]. So, the derivative of pressure with respect to radius is constant which take the
form of following equation
_1 dp = c
r dr
(310)
Intergrating this equation with limits from r to ro, which are the limits encompassing the liquid melt him yields
p{r,5) = ^[r20r2] (3.11)
After solving for the constant by balancing the pressure in z direction, we get
p(r, S) = % [rjj r2]
(3.12)
Now we use Eq. (3.12) with Eq. (3.5) which is the nondimensional variable for pressure to write p(r, z) as
p(r,z) = P(rj)(pvuj) +p(r,8) Now, we apply Eqs. (3.2) (3.4) on Eq. (2.2) to get
(3.13)
1 dp
ru F + ru FH ruzGz = + v
p dr
Fuj ,, /u\ Fuj
b ruF 
r V v / r
(3.14)
By replacing p(r, z) in above equation with Eqs. (3.13) and (3.12) and rearranging, we get
2 7ti2 i 2 771 7 t! 2 /~i2 ^ ^
ru b +roj b H ru G =
p dr
P{puu) + 2^{r2Q r2)
ro
rr fUjy
+ v ruF ( )
L \v S J
(3.15)
After eliminating the common terms on both sides, we obtain
4p0
F + F H G
pr\u2
F
(3.16)
Now we replace the variable F with bb1 from Eq. (3.9) and rearranging terms, we get
(3.17)
H'" = HH" hbL + 2G2 + 2a
Here, a represents the nondimensional external force parameter defined by
4p0
a
pr^uj2
(3.18)
12
Now, to derive the nondimensional magnetic parameter M we employ the same approach as we used above. But in deriving at M we neglect the gradient of pressure with radius in Eq. (2.2) and include the magnetic term aevrB2. We partially rearrange
terms from Eq. (2.2) to get
P
dvr dvr v^ ] dp
by +vz or Oz r
dr
P
d2vr 1 dvr d2vr vr
dr2 r dr dz2 r2
aevrBl (3.19)
Now applying the nondimensional variables Eq. (3.2) (3.4) into the above equation, we get
P
ruj2F2 + ruo2FH' ru2G2
P
Fu fu\ Fu
h ruF 
r V v / r
(jeBlFruj (3.20)
After simplification and eliminating common terms on both sides of equation, we obtain
F2 + F H G2 = 
puj
F + F
We now identify a new magnetic parameter M as
M
puj
(3.21)
(3.22)
So, finally considering the magnetic held and rotation effects, the transformed governing equation become
H'" = HH"  + 2G2 + 2a + MH'
(3.23)
We follow the same set of steps in converting the other governing equations into transformed similarity equations. So from Eq. (2.3)
'Guj ruj2G" Guj~
ru2FG + H{uoj)lBrujG' {pj / u)lB = v
which then becomes
v
OeBo_
P
Gru
crPB2
FG + HG' + FG = N_ Â£ + Q"
puj
After simplifying and rearranging the final transformed governing equation, we get
G" = HG GH + MG
(3.24)
13
Then considering the last component of the momentum equation, i.e Eq. (2.4)
H (uu) l/2H'
v
and rearranging it as
ld_ p dz
P(puu)
9 OO
r2)
(vu)l/2H"
w v 
H{uu)1/2 {yuj )l/2H\u/v)l/2
a; 1/2 P uu
V
+ ()1/2 H"
V V
Hu 3/2/yi/2 = P'u^v1'2 + Wu^v1'2
Hence, after rearranging the final transformed component of the momentum similarity equation becomes
P1 = H" HH'
(3.25)
To transform the energy equation Eq. (2.5), we should define T from Eq. (3.6). So we have T = 9{Tm Tw)+Tw. Hence, applying the transformation on Eq. (2.5), we get
H{izu)l/2d'{Tm Tw) (^1/2) = a(Tm Tw)d"^~ which then becomes
H0\Tm Tw)u = a(Tm TW)Q Simplifying this, we get the similarity energy equation as
6" = PrH6'
(3.26)
Boundary conditions are also transformed based on the nondimensional variable. They are listed below
H' = 0, G = 1, H = 0, e = 0 at p = 0 (3.27)
14
H = 0, G = O' 0=1, at rj = rjs
(3.28)
Ste H(rjs)
Pr 6'(rjs)
(3.29)
Here Eq. (3.29) is the similarity form of Stefan condition, which introduces two additional nondimensional parameters, i.e Stefan number Ste and Prandtl number Pr defined as
where cp is the specific heat of the melt layer
Solving the above equations with finite difference method as discussed in the next section yield the velocity and thermal fields as well as the transport properties at boundaries. In turn, such results help in determining how the rotation of heater and magnetic fields affect the contact melting process.
3.4 Finite Difference Method
Finite difference method is a numerical approach in solving the differential equations, by discretizing the derivatives used in the formulation. The principle behind this approach is the derivatives in the differential equations are substituted with approximations which are derived by linear combination of function values at the grid points. A concrete example of the basic principle of the finite difference method can be seen as follows
Pr
a
v
(3.30)
f('c) = 9
(3.31)
is substituted with
/CE + ft)/M=/,M=g
(3.32)
The computational domain is then divided into multiple grid points. This process
of discretizing the domain is called meshing. In order to achieve accurate results the
15
computational domain is divided into multiple grid points to achieve a fine mesh on the computational domain. These grid points are incremented linearly until the complete computational domain is covered for analysis.
The derivatives in governing equations in the problem statement are replaced with 2nd order approximations at all grid points as specified below. These equations are applied on to all the grid points in computation domain and required results are obtained. For example, the equation
H'" = HH"  + 2G2 + 2a + MH'
2
Hi+2 ~ 2ffi+i + 2Hi_i Hi2 TT Hi+1 2Hi + Hi1 (iA+i Hii)^
2(A q):
and similarity equation
H
(A'qy
2A q
+ 2 G,2 + 2a + M
Hi+1 H
i1
2Aq
(3.33)
G = HG GH + MG becomes after discretization the following
Gi\1 2 Gi + Gi\ Gi+1 Gi\ Hi+\ Hi\
= tLi
Gi + MGi
(3.34)
(Aq)2 1 2A q 2A q
We then convert the above equations into a matrix form to employ the Tridiagonal matrix (TDMA) solution method and determine the results. We can only solve 2nd order equations with the Tridiagonal matrix method and, so the above 3rd order equation is reduced to a 2nd order equation. In this regard, we assume H1 = A, so the equation
Hm HH"   MH' = 2G2 + 2(7
changes to a simpler form given by
A
A" Hi A1 + A~~ MAi = 2G\ + 2 o
(3.35)
16
Now replacing the derivatives with approximations, we get
1 2Ai Aii Hi . a \ a , o
(Kqy 2A~n {A+1 ~ Ai~x) + AY ~ MAl = 2G* + 2(7
and then grouping different terms as coefficients of Vi, Y and Ai+1 as
1 Ht 1 + Ai 2 Ai 1 + M + Ai i r i Hi 1 +
(A q)2 2A q A q)2 2 (A q)2 2A q
2 Gj + 2a (3.36)
For the interior grid points, central differncing approach is employed and near the boundaries, either a forward or a backward differencing scheme is used as appropriate. Similar approach is used for various nonlinear similarity equations, which satisfying the boundary conditions, including the Stefan condition to obtain the numerical solution for a set of characteristic parameters such as a, M, Pr, Ste where the unknown nondimnesional melt layer thickness q$ = Â£(pp2 is obtained as part of the solution.
17
CHAPTER IV
RESULTS AND DISCUSSION
4.1 Introduction
This chapter focus on the results obtained from the physical model considered in the previous chapter. All the variables which are of primary interest to this study are plotted and respective conclusions are derived from them. The parameters which are analysed in this thesis are as follows:
Liquid him velocity held and temperature distribution
Melt him thickness
Melting rate of solid material undergoing phase change
Heat transfer rates at the heater wall and also at the phase change interface location.
Radial shear stress and tangential shear stress, both evaluated at rotating heater wall and also at the phase change interface location.
Following individual sections of this chapter deal with the effect of a particular single parameter and the respective trends obtained graphically are discussed in detail. All the above quantities are studied as a function of the dimensionless external force parameter a and the dimensionless strength of externally applied magnetic held M. In addition, different Prandtl numbers Pr are considered to take into account different huids while changing rotation rates a and magnetic helds M. Also the Stefan condition helps us to interpret the effect of phase change phenomenon in melting process. Hence, the inhuence of phase change process is included in the results in the form of changing Stefan numbers Ste. Each section is divided into two subsections and each of the subsection deals with a particular variable changing with a and changing with M.
18
4.2 Velocity and Temperature profiles within Melt Layer
When a liquid melt layer is formed during the melting process, external force a and external magnetic held strength M significantly affect its velocity and temperature distributions. Study of these velocity components in the radial, axial and angular direction is important in the analysis of heat transfer and melting rates. In the following subsections, we discuss in detail the effect of a and M on the radial, axial and angular velocity components for various values of Stefan and Prandtl numbers. These figures are plotted as a function of the dimensionless axial coordinate rj on the abscissa and a dimensionless velocity component on the ordinate. It is observed that when changing a or M, the maximum possible values of rj change on the abscissa in these figures indicating the change in the thickness value of the melt layer for that particular value of a and M.
4.2.1 Radial velocity component
Velocity component of the liquid melt layer along the radial direction is defined as radial velocity. It is converted into a nondimensional variable F from Eq. (3.2) in Chapter 3.
19
4.2.1.1 Effect of external force on radial velocity of the melt layer
v
(a)
(b)
20
V
(c)
(d)
Figure 4.1: Variation of radial velocity of the liquid melt layer as a function of
external force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
21
Figures 4.1 (a) (d) show the variation of F vs q as a function of a for different Pr. It can be seen that the radial velocity increases with increasing external load parameter a. With an increase in
4.2.1.2 Effect of magnetic field on radial velocity of the melt layer
(a)
22
V
(b)
v
23
0.012
(d)
Figure 4.2: Variation of radial velocity of the liquid melt layer as a function of magnetic field parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
By contrast, as shown in Figs. 4.2 (a) (d), the effect of magnetic held represented by M is seen to reduce the magnitude of the radial velocity component. It is evident that as the magnetic held strength increases it decreases the velocity values due to the retarding effect of the additional forces arising from the magnetic held effects i.e Lorentz forces. These Lorentz forces act in the opposite direction of radial velocity. Furthermore, it is found that there is a decrease in the velocity magnitudes with an increase in the Prandtl number. Interestingly, for lower Prandtl number huids at higher magnetic helds there is a exceptional increase in the maximum value of i] which has broadened the velocity prohles due to a relatively thicker melt layer, which can be seen in Fig. 4.2 (a).
4.2.2 Axial velocity component
Velocity of melt layer component defined along the direction of the central axis
which passes through melt and heater is defined as the axial velocity. It is also
24
converted into a nondimensional variable H from Eq. (3.3). The quantity H in these figures indicate its flow direction opposite to the coordinate axis.
4.2.2.1 Effect of external force on axial velocity of the melt layer
v
(a)
v
(b)
25
V
(c)
(d)
Figure 4.3: Variation of axial velocity of the liquid melt layer as a function of
external force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
26
Figures. 4.3 (a) (d) show the effect of a on the axial velocity profile for different values of Pr. It is seen that increasing a has a similar effect on the axial velocity profiles by increasing the melt flow generated due to higher external loads and centrifugal forces during contact melting. The slopes of H are found to increase with increase in a. The maximum values of q i.e melt layer thickness are also seen to decrease with increase in a. Also, as before, increasing Prandtl number decreases the axial velocity. For lower Prandtl number fluids with higher values of
4.2.2.2 Effect of magnetic field on axial velocity of the melt layer
(a)
27
V
(b)
v
28
(d)
Figure 4.4: Variation of axial velocity of the liquid melt layer as a function of magnetic field parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
On the other hand, increasing magnetic fields decreases the axial velocity magnitudes because of the retarding Lorentz forces acting on the liquid melt layer seen in Figs. 4.4 (a) (d). This decrease in the velocity magnitudes are accompanied by an increase in the melt him thickness. In addition, such a decrease in velocity is further intensified with increase in the Prandtl number.
4.2.3 Angular velocity component
Velocity defined along the circumferential direction of the rotating heater is defined as angular velocity component. After nondimensionalization it is rewritten in terms of the variable G.
29
4.2.3.1 Effect of external force on angular velocity of the melt layer
v
(a)
(b)
30
V
(c)
11
(d)
Figure 4.5: Variation of angular velocity of the liquid melt layer as a function of external force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
31
Effect of external load parameter a on the angular velocity profiles for different Pr are shown in Figs. 4.5 (a) (d). Increasing a increases slope of velocity profiles of G. But when observing for a particular value of ?/, increase in a decreases the velocity magnitudes which is different from other velocity profiles. This is as a result of the mass conservation in the melt layer. To balance the melt layer mass coming into and going out of system there is a corresponding decrease in the angular velocity. Slope of the graphs are also seen to decrease with increase in Prandtl numbers. Higher Prandtl number fluids velocity profiles seem to be straight lines with a constant slope when compared with lower Prandtl number fluid cases.
4.2.3.2 Effect of magnetic fields on angular velocity of the melt layer
(a)
32
(b)
v
(c)
33
(d)
Figure 4.6: Variation of angular velocity of the liquid melt layer as a function of magnetic field parameter M for different Prandtl numbers Pr when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
Increasing the magnetic held strengths is seen to decrease the magnitudes of the velocity profiles and also increase their slopes (see Figs. 4.6 (a) (d)). The effect of magnetic held is found to be more pronounced for lower Prandtl number huids rather than for higher Prandtl number huids. Looking at Figs. 4.6 (a) (d) it is seen that at higher Prandtl number, variation between the velocity prohles are found to be insignificant.
4.2.4 Temperature distribution
Study of temperature distribution within the melt formed is important to understand the heat transfer rates between heater, melt and melting solid. After nondimensionalizing temperature is represented with nondimensional variable 9.
34
4.2.4.1 Effect of external force on temperature distribution of melt layer
v
(a)
v
(b)
35
V
(c)
11
(d)
Figure 4.7: Variation of the temperature of the liquid melt layer as a function of external force parameter a for different Prandtl numbers Pr when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
36
As shown in Figs. 4.7(a) (d), temperature profiles within the melt layer show a constant increase in their slopes with increase in the external load parameter a. In addition, the thickness of the melt layer accompanying such a variation in temperature is found to decrease. Increasing the Prandtl number also decreases melt layer thickness.
4.2.4.2 Effect of magnetic fields on temperature distribution of the melt layer
(a)
37
V
(b)
(c)
38
(d)
Figure 4.8: Variation of the temperature of the liquid melt layer as a function of magnetic held parameter M for different Prandtl numbers Ste when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
Increasing the magnetic parameter M decreases the slopes of the temperature profiles (see Figs. 4.8 (a) (d)). Highest slopes for the temperature is registered at M = 0. At higher Prandtl numbers, there is not appreciable difference in the temperature profiles with changing external magnetic fields M.
4.3 Melt Film Thickness
As discussed earlier the melting solid is separated from rotating heater with a thin layer of melt developed in between them. So as the melting progresses the thickness of the liquid melt increases and heater is separated from melting solid by a thickness which becomes a constant at steady state. We consider the external force parameter a taking on values of 0, 0.5, 1 and 5, the magnetic parameter M as 0, 0.5, 1, 3 and 5, the Stefan number Ste as 0, 0.05, 0.1, 0.5 and 1, and Prandtl number Pr as 1, 10,
39
50, 100. We then consider the variation of dimensionless melt layer thickness i]s as either a function of a or M by keeping the other parameter as constant.
4.3.1 Effect of external force on melt the layer thickness
(a)
(b)
40
0.8
(c)
(d)
Figure 4.9: Variation of the thickness of the liquid melt layer as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
41
Figures. 4.9 (a) (d) presents the variation of the melt layer thickness i]s as a function of the external force parameter a. This figure also represents the effect of changing the Stefan number as a parameter for various values of Prandtl number on the melt layer thickness. It is evident that the thickness of the melt is reduced as a increases for a given Stefan number, while by contrast it is increased as Stefan number increases for a give a. Also higher Prandtl fluids retain thinner layers. The magnitude of the variation of i]s or its slope is higher for lower values of a and for higher values of the Stefan number. As Prandtl number decreases, a sharp decrease in magnitudes of the thickness is observed at high Stefan numbers and lower a. Increasing external force parameter a facilitates greater rate of melt removal, which is formed in between the rotating heater and melting solid and hence the layer becomes thinner. And increase in Stefan number indicates addition of more sensible heat to system relative to latent heat, while the decrease in Prandtl number indicates the greater role of the thermal diffusivity, both of which cause the melt layer to become thicker.
Figure 4.10: Variation of thickness of the liquid melt layer as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter
M = 5 and Pr=10.
42
Figure 4.10 is represents the combined effect of the external force parameter a and the magnetic parameter M, which is set to be M = 5. So when comparing with Fig 4.9 (b) which is for M = 0 (no magnetic field case), we see increased values of thickness because of the additional retarding magnetic forces acting on the melt, which inhibits its removal. It is noticed that the trend of Stefan number is similar even under the combined effect of a and M but there is significant increase in the magnitude of thickness when compared to the case without a magnetic field.
4.3.2 Effect of the the magnetic field on the melt layer thickness
(a)
43
2.5
Pr = 10 a = 0
S? 1.5
Ste = 1
Ste = 0.5
0.5
y Ste = 0.1
Ste = 0.05
) & O
0.5 1 1.5 2 2.5 3 3.5 4 4.5
M
(b)
M
(c)
44
(d)
Figure 4.11: Variation of thickness of the liquid melt layer as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter a = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
Figures 4.11 (a) (d) show effect of changing the external magnetic hied parameter M on the melt him thickness for different sets of Pr, by keeping Ste as a constant parameter in each case. The variations in i]s initiated by increasing magnetic held is different from increasing the external force. It is seen that the melt layer becomes thicker with increase in the magnetic number M, which is contrasting with the behaviour observed when the external force parameter is increased. And also an increase in the Stefan number and a decrease in the Prandtl number is seen to increases him thickness formed between heater plate and melting solid. When observing Fig 4.11 (a), it seen that for low Prandtl number hows there is a sharp increase in melt him thickness. For example, the increase in thickness when the magnetic number changes from 3 to 5 is a factor of 2 for a particular value of the Stefan number for lower
45
Prandtl number fluids when compared to higher Prandtl number fluids. So, for lower Prandtl number fluids, thicker melt films are developed with higher magnetic fields when coupled with higher Stefan numbers. This increasing trend in thickness for higher M is because of the additional opposing Lorentz forces which act on the melt when subjected to external magnetic fields. Thus, higher the magnetic number M, larger the magnitude of retarding magnetic forces acting on melt film thickness.
4.4 Melting Rate
Melting rate is defined as the amount of melting solid getting converted to melt per unit time at the interface. This can de found by looking at the melt velocity pattern in axial direction along the central axis of rotating heater. This melt velocity need to be recorded at the phase change interface location. Hence the figures in this section depict the nondimensional melting velocity H(i]s) on the ordinate and varying a or M on the abscissa.
4.4.1 Effect of external force on melting rate
(a)
46
0.15
(b)
(c)
(d)
Figure 4.12: Variation of the melting rate as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l,
(b)Pr=10, (c)Pr=50, (d)Pr=100.
Figures 4.12 (a) (d) show the effect of the external force parameter a and Stefan Number on melting rate for various sets of Pr. As either the value of a or Stefan number Ste increases, melting rate of the solid block also increases. But as the Prandtl number increases, by contrast the melting rate decreases. Also at higher a, the change in the melting rate when Stefan number varies from 0.5 to 1 is larger for Prandtl number 1. The same increase in melting rate is not observed for Prandtl number 100 in Fig. 4.12 (d). In general, higher values of a corresponds to greater external loads, which enhances the contact melting process. Moreover, when the role of the specific heat relative to the latent heat is more pronounces as reflected in higher Ste, it induces melting in the solid at a more rapid rate By contrast, when the melt layer has greater role for the viscous diffusion effects (i.e high Pr) its removal
48
is inhibited, which in turn, slows down the melt generation process.
Figure 4.13: Variation of the melting rate as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 5 and Pr=10.
The combined effect of a and M on melting rate is shown in Fig. 4.13. It is clear that, there is a small decrease in the melting rate because of the presence of magnetic fields when compared to the M = 0 case (see Fig. 4.12(b)).
4.4.2 Effect of magnetic field on melting rate
Figures 4.14 (a) (d) show the effect of magnetic fields applied on to liquid melting rate for different sets for Pr and Ste. It is seen that the melt rate decreases with an increase in the magnetic parameter M. In a similar fashion as before, for higher values of the Stefan number and lower values of the Prandtl number, high melting rates are registered. Moreover, for lower Prandtl number fluids, the effect of magnetic held is more pronounced in reducing the melting rate for high values of the Stefan number. When comparing Fig. 4.14 (a) and (d), for the same value of the Stefan number, the variation is higher for lower Pr fluids compared to high Pr fluids.
49
M
(a)
M
(b)
50
0.022
Ste = 0.1
Ste = 0.05 <
0.5 1 1.5 2 2.5 3 3.5 4 4.5
M
(c)
(d)
Figure 4.14: Variation of the melting rate as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter a = 0. (a)Pr=l,
(b)Pr=10, (c)Pr=50, (d)Pr=100.
51
4.5 Heat Transfer Rate
Heat transfer rate is an important parameter when determining the effect of magnetic fields and external force parameter on direct contact melting. Various modes of heat transfer which generally occur in contact melting are heat conduction and heat convection. Heat conduction mode generally dominates in contact melting undergoing rotation [2] because of the relatively small melt him thickness between the melting solid and the heater. There can also be some heat convection as the liquid melt layer is formed in between heater and PCM and is squeezed out of system undergoing rotation. Heat transfer rates are studied at two different locations in this thesis; the first location is defined at the rotating heater wall and second location is defined at the phase change interface location. Studying both locations are critical in understanding the rotation and magnetic effects on the thermal transport. Heat transfer rate at the heater wall provides information about heat exchange between rotating heater wall and adjacent liquid melt formed besides heater wall which in turn provides the heat input to initiate melting on the other side. Heat transfer rate at the phase change interface location provides the heat exchange between liquid melt and melting solid where there is additional latent heat effect to be considered. Heat transfer rate is usually is studied with Nusselt number Nu = ^ After applying transformation Nusselt number can be rewritten in terms of nondimensional temperature gradient ^(O) Hence, the figures that follow represent variation between ^ (O) and either a or M. It may be noted that O' (0) indicates the heat transfer rate at the heater wall while 0'(r]s) represent that at the phase change location.
52
4.5.1 Effect of external force on heat transfer rate at the heater wall
(b)
53
6
(c)
(d)
Figure 4.15: Variation of the heat transfer rate evaluated at heater wall as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0. (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100.
54
Figures 4.15 (a) (d) represents the effect of the external force parameter a on the heat transfer rate at the heater wall for different sets of Ste and Pr. Clearly increasing trend in the heat transfer rates at wall is observed with increase in a as well as increase in Pr. By contrast the heat transfer rate decreases with increase in Ste. Such a later trend with Ste was not observed in our previous analysis of the variation of the melt film thickness or the melting rate. Increasing trend of heat transfer rate with increase in the Prandtl number can be readily related to the concepts of the boundary layer theory. As the melting process progresses, thermal boundary layer and viscous boundary layers are developed. The heat transfer rate is related to the thermal boundary layer thickness. Based on a scaling, an increase in Prandtl number indicates decrease in thermal boundary layer thickness
~s~ Pr 1/3 [17]. Then, the relation between thermal boundary layer and heat transfer rate coefficient can be inferred from h = jS. Hence, we find an increase in the heat transfer rate with increase in Prandtl number. The enhancement in the wall heat transfer rate due to the external forcing parameter a can be explained as follows.
As a increases, it increases the squeezing flows rates within the melt layer as noticed in earlier sections. Such higher flow rates within the liquid film would then carry away greater quantities of thermal energy from the heater wall. On the other hand a decreasing in O' (0) with Stefan number is noticed, because the larger value of Ste, greater portion of energy is contributed towards the specific heat effects in the solid within the solid rather than redistributing it to the convecting heating within the liquid film.
55
4.5.2 Effect of external force on heat transfer rate at the phase change location
(a)
(b)
56
(c)
(d)
Figure 4.16: Variation of heat transfer rates evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
57
Figures 4.16 (a) (d) represents the influence of the external force parameter a on the heat transfer rate at the interface for different sets of Ste and Pr. It can be seen that the heat Transfer rate at the phase change location is also increasing with increase in either a or Pr. As before, it can be observed that increasing the Stefan number decreases the heat transfer rate at the phase change location. These trends can be explained in an analogous manner as done for the heater wall. Let us now see the differences between the heat transfer rate predictions at the wall and the interface. For example, consider for each case when Pr = 100. Now, when compared with the heat transfer rate at wall location there is a 2% decrease in the heat transfer rate at the interface for Ste = 0.05, 5% decrease for Ste = 0.1, and 20% decrease for Ste = 0.5 and a large 32% decrease for Ste = 1. Such differences between O' (0) and O' ()]s) with Ste can be attributed to the increasing portion of the thermal energy being stored in the solid due to specific heat effects.
4.5.3 Effect of magnetic field on heat transfer rate at the heater wall
(a)
58
M
(b)
(c)
59
(d)
Figure 4.17: Variation of heat transfer rates evaluated at heater wall as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter a = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
Figures 4.17 (a) (d) represent the effect of magnetic held on the heat transfer rate at the heater wall. It is seen that the heat transfer rate decreases with increase in magnetic fields. Such a decrease is seen to be more pronounced with increase in Stefan number. The reduction in heat transfer rates with M is associated with reduced melting rates and lower squeeze him how arising due to the Lorentz forces, as discussed in earlier sections.
60
4.5.4 Effect of magnetic fields on heat transfer rates at the phase change location
M
(a)
M
(b)
61
3
2.5
Ste = 0.05
e
Pr = 50 a = 0
o
3
1.5
Ste = 0.1

0.5'''''''''
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
M
(c)
(d)
Figure 4.18: Variation of heat transfer rates evaluated at phase change location as a
function of magnetic parameter M for different Stefan numbers Ste when external force parameter a = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
62
As in the previous case, the heat transfer rate at the interface 9' (r/s) decreases with an increase in the magnetic parameter M (see Figs. 4.f8 (a) (d)). Such an effect is seen to be more pronounced at lower Pr. For example, when Pr = 1, there is a 43% reduction in the heat transfer rate at the interface for the case with M = 5, when compared to the no magnetic held case (M = 0). Flowever, the reduction becomes less than 4% when Pr = 100 for similar cases of M. Also, as comparisons between O' (0) and 0'(rjs), there is about 34% decrease for the 0'(rjs) case when Ste = 1, 5% decrease when Ste = 0.1 and 2.6% reduction for Ste = 0.05.
4.6 Tangential Shear Stress and Radial Shear Stress
We now discuss how the various components of the shear stress at the heater wall and the interface location vary with various characteristic parameter. The radial shear stress is a shear stress acting in radial direction on the rotating heater On the other hand, the tangential shear stress indicates the amount of torque required to keep the rotating heater in circular motion. Both components of the shear stress can be derived from the Newtonian shear expression ly, = v j for the tangential shear
stress and rr = v for the radial shear stress [13]. We apply our similarity
transformations on these equations to convert them to appropriate nondimensional variables. Flence the tangential shear stress becomes G"(0) at the heater wall and G'(rjs) at the interface location. Radial shear stress take the form H"(0) at the heater wall and H"(r)s) at the phase change location. Plots for the shear stress components are shown between G'(0) and G'(r)s) c>n the ordinate with either a or M on abscissa; similarly H"(0) and H"(r)s) on the ordinate as a function of either a or ill on abscissa.
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4.6.1 Tangential shear stress
4.6.1.1 Effect of external force on tangential shear stress at the heater wall
(a)
(b)
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(c)
(d)
Figure 4.19: Variation of tangential shear stress evaluated at wall as a function of
external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
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Figures 4.19 (a) (d) shows the effect of the external force parameter a on tangential shear stress G'(0) at the heater wall for various sets of Pr and Ste. It is evident that an increase in a results in increase in the tangential stress. By contrast, for a given
4.6.1.2 Effect of external force on tangential shear stress at the phase change location
(a)
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(b)
(c)
67
7
(d)
Figure 4.20: Variation of tangential shear stress evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100)
As shown i n Figs. 4.20 (a) (d), the tangential shear stress at the interface location increases with an increase in the external force parameter a. Similar to that at the heater wall, the tangential stress at the interface location shows similar trends with an increase in Prandtl number and a decrease in the Stefan number. It may be noted that the difference in the magnitudes of the shear stress at the wall and interface location are negligible, as G'(i]s) is on an average only 0.7% smaller when compared to G'(0).
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4.6.1.3 Effect of magnetic field on tangential shear stress at the heater wall
M
(a)
M
(b)
69
3.2
(c)
(d)
Figure 4.21: Variation of tangential shear stress evaluated at wall as a function of
magnetic parameter M for different Stefan numbers Ste when external force parameter Ste = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100)
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Figures 4.21 (a) (d) presents the influence of the magnetic parameter M on the tangential stress at the wall G'(0) for different sets of Ste and Pr. An increasing trend in the tangential shear stress is observed with increasing magnetic fields. It is also found to increase with decrease in Stefan number and increase in the Prandtl number like in the earlier case. Increase in tangential stress with increase in magnetic fields can be explained with equation ly, = v + 4^ j. Tangential shear stress increases with increase in the gradient of V^, when magnetic field becomes stronger, as can be seen from the velocity profiles discussed earlier. It is also found that the shear stress has a more pronounced variation with M at higher Ste and at lower Pr.
4.6.1.4 Effect of magnetic field on tangential shear stress at the phase change location
(a)
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M
(b)
M
(c)
72
(d)
Figure 4.22: Variation of tangential shear stress evaluated at phase change location as a function of magnetic parameter M for different Stefan numbers Ste when external force parameter Ste = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100)
Figures 4.22 (a) (d) illustrates the effect of magnetic held on the tangential shear stress at the interface for various sets of Ste and Pr. Interestingly there is a change in tangential shear stress patterns at the phase change location as a function of magnetic held. There is a decreasing trend at the interface, when compared to increasing trend at heater wall location. Effect of the reduced velocity gradients with M at the phase change location and the melt how generation is anticipated to be the main cause of this change in trend. This change in trend needs further investigation, to understand the different behaviour of the tangential stress and the underlying physics behind it.
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4.6.2 Radial shear stress
4.6.2.1 Effect of external force on radial shear stress at the heater wall
(b)
74
(c)
(d)
Figure 4.23: Variation of radial shear stress evaluated at wall as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0, (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100)
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The variation of radial shear stress at the wall H"(0) with the external force parameter a for different sets of Ste and Pr are presented in Figs 4.23 (a) (d). It can be noticed that there is a steady increase in the radial shear stress measured at the heater wall with increase in a as well as due to Ste. As a increases, the external pressure force acting on melting solid increases causing increased radial squeezing flow and accompanied by higher radial velocity gradients. Hence, there increasing trend in the radial shear stress is registered with a. At higher a, the variations in the radial shear stress with Ste also increases. For example, when Pr = 100, at a = 0, as ste changes from 0.5 to 1.0, the variation in H"{0) is only 0.6%, while at a = 5, for similar changes in Ste, the variation in H"(0) takes a higher value at 8%. Also it can be seen as the Prandtl number increases, the radial shear stress at wall decreases.
4.6.2.2 Effect of external force on radial shear stress at the phase change location
(a)
76
(b)
(c)
77
(d)
Figure 4.24: Variation of radial shear stress evaluated at phase change location as a function of external force parameter a for different Stefan numbers Ste when magnetic parameter M = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
Figures 4.24 (a) (b) show the variation in the radial shear stress evaluated at the phase change location. It can be seen that as a increases for a given Ste or, conversely, as Ste increases for a given a, the radial shear stress at the interface increases. By contrast, the radial shear stress decreases with an increase in Pr. These results are very similar in trend with that analysed for the heater wall in the previous section, except that there is a sign change. Looking closely at radial stress results evaluated at the heater wall, the respective figures are plotted between H"(0) and a. But when evaluated at the phase change location graphs are plotted between H"{0) and a. In general radial shear stress is changing from negative values to positive values when studied over the entire length of the melt him thickness.
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4.6.2.3 Effect of magnetic field on the radial shear stress at the heater wall
M
(a)
M
(b)
79
(c)
(d)
Figure 4.25: Variation of radial shear stress evaluated at heater wall as a function of magnetic fields M for different Stefan numbers Ste when external force parameter <7 = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
80
The effect of magnetic field parameter M on the radial shear stress at the heater wall H"(0) for different sets of Ste and Pr is shown in Figs. 4.25 (a) (d). A decreasing trend in the radial shear stress is observed with increase in M and decrease in Ste. From the equation rr = v + ^f) it is seen that the radial shear stress increases with increase in gradient of Vr. Increasing the magnetic field strength decreases the slope of the velocity profile in radial direction as seen from Figs. 4.3 (a) (d). This leads to a decreasing trend in the radial stress H"(0) as observed in Figs. 4.25 (a) (d). In addition decreasing radial shear stress seems to be dominant at higher Stefan number especially for lower Prandtl number fluids. As Prandtl number increases values of stress decreases. For lower Prandtl number flows degree of decreasing slope at higher Stefan number is more when comparing to lower Stefan numbers. It is also seen that as Prandtl number decreases there is a reversal in the stress value trend as shown in Fig. 4.26 (a).
4.6.2.4 Effect of magnetic field on radial shear stress at the phase change location
(a)
81
M
(b)
M
82
0.11
0.04'''''''''
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
M
(d)
Figure 4.26: Variation of radial shear stress evaluated at phase change location as a function of magnetic fields M for different Stefan numbers Ste when external force parameter a = 0 (a)Pr=l, (b)Pr=10, (c)Pr=50, (d)Pr=100
Figures 4.26 (a) (d) show the influence of M on the radial stress at the interface H"(i]s) for different sets of Ste and Pr. it is evident that as M increase, the radial stress at the interface decreases. Flowever, the effect of the Ste on H"(i]s) exhibits a complex trend due to the interaction of melt flow generation and Lorentz force effects. On an average, about 34% decrease in H"(i]s) are recorded with increase in M from 0 to 5 for the Pr = 100 case.
Furthermore, it can be seen that the magnitude of radial stress decreasing with increase in Prandtl number at the phase change location. Also, as observed in the previous section, for lower Prandtl number fluids a drastic decrease in is
observed for higher Stefan numbers. At higher M and lower Pr the radial stress values are decreasing with increase in the Stefan number. In fact, the stress values at
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M = 5 for the Stefan numbers 0.5 and f are much lesser that for the Stefan number of 0.05 and 0.1.
When analysing the radial stress with changing a, we have noted the radial stress changes its value from ve to +ve from the heater wall to phase change location. Similar pattern is observed when studying the radial stress for variations with external magnetic fields.
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CHAPTER V
SUMMARY AND CONCLUSION
This chapter deals with conclusions which are deduced from the my thesis. In this thesis, a physical model for the closed contact melting process undergoing rotation in the presence of a magnetic held is developed The formulation is based on three dimensional mass, momentum and energy equations along with the Stefan condition at the interface and subjected to Lorentz force, which are written in the cylindrical coordinate system. They are reduced to a simpler similarity formulation by means of similarity transformation, which facilitate numerical solution by means of a finite difference method. The transport physics of this rotating contact melting problem are governed by a set of characteristic dimensionless parameter including the external force parameter a, magnetic parameter M, Stefan number Ste and Prandtl number Pr. Numerical results are obtained to systematically study the effects of the parameter on the velocity profiles structures within the liquid melt layer, melting rates, heat transfer rates and shear stress at the heater wall and the interface. Following are the main finding in this study.
The radial and axial velocity components of the melt layer are seen to increase with increase in external forcing parameters and decrease with Prandtl number Pr. Only the angular velocity component of melt layer is found to decrease with increase in a to maintain a mass balance in the rotating contact melting system. But increasing magnetic held parameter M decreases the magnitudes of the velocity in all directions. In addition increase in the temperature values with increase in a is noticed. On the other hand, the application of the magnetic held with increasing strengths is seen to cause a decrease in the magnitudes of the temperature prohles with the melt layer.
When observing the melt him thickness variations, the thickness is found to decrease with an increase in a. This is because as a increases, the centrifugal force acting on the melt him increases and hence it accommodates more melt removal from
85
the system. Also as Stefan number increases, which results in a greater portion of the heat supply going to raise the specific heat of the melting solid, the melt him becomes thicker. Greater variations in the melt layer thickness for lower a and higher Stefan number is observed at lower Prandtl number because of more prominent role of thermal diffusivity effects. When studying the effects of magnetic fields, the melt layer thickness is seen to increase with increase in M because of additional retarding Lorentz forces acting on the liquid melt. An important finding which can be deduced is that increasing the magnetic held strength reduces the magnitudes of melt velocities and heat transfer rates. These are analysed in more detail following sections.
We now identify that a, Ste and Pr have similar effects on the melting rates as they were on the melt layer thickness. Increasing a or Ste enhances the transfer of heat from the rotating heater into the melting solid. This is again more pronounced for lower Pr fluids. Hence decreasing trend in the melt him thickness and increasing increasing variations of the melting rate can be correlated with each other for rotating contact melting system.
Next we summarize the main deductions based on the heat transfer rate results. In general it is known that increasing the Prandtl number decreases thermal boundary layer thickness. As the thermal boundary layer decreases, it increase heat transfer rate. Thus higher Pr fluids generated during the contact melting process is accompanied by higher heat transfer rates at both the heater wall and the interface. Another conclusion which can be deduced as, at lower values of a convection heat transfer becomes significant in the liquid melt. On the other hand, as a as increases, the thickness of melt layer decreases and heat conduction becomes more dominant. Increase in the Stefan number indicates greater amount of sensible heat being added to system. Hence, the difference in the heat transfer rate at the heater wall and phase
86
change location is relatively large of about 32% for higher values of Stefan number.
Tangential shear stress generated due to the liquid melt flow is found to increase with increase in external force parameter a and increase in the magnetic parameter M. Both these parameters add additional torque to the rotating heater, which results in an increase in shear stress. But they are found to decrease when evaluated at phase change location in the case of magnetic held strengths M.
Another quantity of interest is the radial shear stress. It is observed that the radial shear stress increases with increase in the external force parameter a. On the other hand, the radial shear stress is seen to decrease with increase in M. In general, these conclusions are consistent with the fact that higher values of external force parameter promotes greater melt removal rates, while by contrast higher magnetic held strengths is seen to decrease the magnitudes of melt layer velocities.
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INFLUENCEOFMAGNETICFIELDONDIRECTCONTACTMELTING UNDERGOINGROTATION by SRIKESHMALEPATI B.Tech,GitamUniversity,2014 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience MechanicalEngineeringProgram 2016
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ThisthesisfortheMasterofSciencedegreeby SrikeshMalepati hasbeenapprovedforthe MechanicalEngineeringProgram by KannanPremnath,Chair SamuelWelch DanaCarpenter December17,2016 ii
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Malepati,SrikeshM.S.,MechanicalEngineering InuenceofMagneticFieldonDirectContactMeltingUndergoingRotation ThesisdirectedbyAssistantProfessorKannanPremnath ABSTRACT Contactmeltingheattransferisamajorindustrialprocesswhichhasdiverse applicationsinvariousengineeringelds.Itisaphysicalprocessachievedbyforced contactbetweenaheatingsourceandaphasechangematerialundergoingmelting. Contactmeltingprocessesareclassiedbasedonmanyfactorsliketherelativemotionbetweenheatingmaterialandmeltingsubstance,locationofthephasechange interfaceandmotionofheatingsubstanceetc.Previouslytherehavebeennumerous studiesonvariousfactorswhichaectthemeltingrateincontactmeltingprocess. Magnetohydrodynamicshasgainedimportanceinrecentyearsintheengineeringeld. Theeectofmagneticforcesoncontactmeltingisaninterestingareaofresearch. Thisthesisdealswithdeterminingtheeectsofmagneticeldsonclosecontactmeltingprocessessubjectedtorotation.Computationaldomaininthisworkisdescribed asameltingsubstanceplacedonrotatingheaterandissubjectedtoexternalmagnetic elds.Governingequationsaredevelopedforthisproblembyreferringtopreviousliterature,andadditionalforcesinducedbecauseofthemagneticeldsareindicatedas additionaltermsinthegoverningequations.Variouscharacteristicnondimensional variablesaredenedandallgoverningequationsinvolvingthreedimensionalmass, momentumandenergyequationsincylindricalcoordinatesystemcoupledwithan interfacialtransportequationaretransformedintosuchnondimensionalvariables leadingtoasetofsimilarityequations.Finitedierencemethodsareemployedin solvingthetransformedgoverningequations.Externallyspeciedvariableslikerate ofcircularrotationandstrengthofmagneticeldhavealsobeennondimentionalized andarereferredtoasexternalforceparameter andmagneticparameter M .Nuiii
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mericalresultsarethenobtainedfromthesolutionofthesimilarityequationsand theeectofvariousgoverningnondimensionalvariablesonthecontactmeltingand heatprocessesundergoingrotationinthepresenceofamagneticeldarestudied systematicallyandinterpreted.Inparticularourresultsrevealthatmeltingrateand heattransferrateincreasewithincreasein ,whilesuchratesdecreasewithincrease in M Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:KannanPremnath iv
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ThisthesisisdedicatedtoALMIGHTYSHIVA,toMYPROFESSORandto MYFAMILY. v
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ACKNOWLEDGMENTS FirstlyIwouldliketothankmyprofessorandadvisorDrKannanPremnath fordeningmythesisproblemstatementandguidingmeinsolvingit.Withouthis constantsupportthiswouldnotbepossible.IwouldliketoalsothankDrSamuel WelchandDrDanaCarpenterforagreeingtobeapartofmythesiscommittee. VeryspecialandsincerethankstoFarzanehHajabdollahiforhelpingmeunderstand,developandmodifytheMATLABcodetosolvemythesisproblemstatement. IwishtothankBhanuBabaiahgariforhisassistancewithLaTexandothercomputingissues. IwouldalsoliketoappreciateKennethSiscoandhisteamfromengineeringcomputerlabforprovidingmeallkindsofcomputingservices. SpecialthankstomyFAMILYandFRIENDSforsupportingmethroughoutmy mastersprogram. vi
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TABLEOFCONTENTS CHAPTER I.INTRODUCTION...............................1 1.1Introduction...............................1 1.2LiteratureReviewonClosecontactmelting..............1 1.3LiteratureReviewonCloseContactMeltingwithRotatingDisk..2 1.4LiteratureReviewonMagnetohydrodynamics.............2 1.5PresentWork..............................3 II.MATHEMATICALMODELLING.......................5 2.1Introduction...............................5 2.2ComputationalDomainDescription..................5 2.3MagneticEects.............................7 2.4GoverningEquations..........................8 III.NONDIMENSIONALPARAMETERS...................10 3.1Introduction...............................10 3.2NonDimensionalVariable........................10 3.3TransformedGoverningEquations...................11 3.4FiniteDierenceMethod........................15 IV.RESULTSANDDISCUSSION........................18 4.1Introduction...............................18 4.2VelocityandTemperatureproleswithinMeltLayer.........19 4.2.1Radialvelocitycomponent....................19 4.2.1.1Eectofexternalforceonradialvelocityofthemeltlayer20 4.2.1.2Eectofmagneticeldonradialvelocityofthemeltlayer22 4.2.2Axialvelocitycomponent....................24 4.2.2.1Eectofexternalforceonaxialvelocityofthemeltlayer25 4.2.2.2Eectofmagneticeldonaxialvelocityofthemeltlayer27 vii
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4.2.3Angularvelocitycomponent...................29 4.2.3.1Eectofexternalforceonangularvelocityofthemelt layer............................30 4.2.3.2Eectofmagneticeldsonangularvelocityofthemelt layer............................32 4.2.4Temperaturedistribution....................34 4.2.4.1Eectofexternalforceontemperaturedistributionof meltlayer.........................35 4.2.4.2Eectofmagneticeldsontemperaturedistributionof themeltlayer.......................37 4.3MeltFilmThickness...........................39 4.3.1Eectofexternalforceonmeltthelayerthickness......40 4.3.2Eectofthethemagneticeldonthemeltlayerthickness..43 4.4MeltingRate...............................46 4.4.1Eectofexternalforceonmeltingrate.............46 4.4.2Eectofmagneticeldonmeltingrate.............49 4.5HeatTransferRate...........................52 4.5.1Eectofexternalforceonheattransferrateattheheaterwall53 4.5.2Eectofexternalforceonheattransferrateatthephasechange location..............................56 4.5.3Eectofmagneticeldonheattransferrateattheheaterwall58 4.5.4Eectofmagneticeldsonheattransferratesatthephase changelocation..........................61 4.6TangentialShearStressandRadialShearStress...........63 4.6.1Tangentialshearstress......................64 4.6.1.1Eectofexternalforceontangentialshearstressatthe heaterwall.........................64 viii
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4.6.1.2Eectofexternalforceontangentialshearstressatthe phasechangelocation...................66 4.6.1.3Eectofmagneticeldontangentialshearstressatthe heaterwall.........................69 4.6.1.4Eectofmagneticeldontangentialshearstressatthe phasechangelocation...................71 4.6.2Radialshearstress........................74 4.6.2.1Eectofexternalforceonradialshearstressatthe heaterwall.........................74 4.6.2.2Eectofexternalforceonradialshearstressatthephase changelocation......................76 4.6.2.3Eectofmagneticeldontheradialshearstressatthe heaterwall.........................79 4.6.2.4Eectofmagneticeldonradialshearstressatthe phasechangelocation...................81 V.SUMMARYANDCONCLUSION.......................85 REFERENCES...................................88 ix
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FIGURES 2.1Computationaldomainforthecontactmeltingproblemundergoingrotationinthepresenceofmagneticeld....................7 4.1Variationofradialvelocityoftheliquidmeltlayerasafunctionofexternalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......21 4.2Variationofradialvelocityoftheliquidmeltlayerasafunctionofmagneticeldparameter M fordierentPrandtlnumbers Pr whenexternal forceparameter =0.aPr=1,bPr=10,cPr=50,dPr=100.....24 4.3Variationofaxialvelocityoftheliquidmeltlayerasafunctionofexternalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......26 4.4Variationofaxialvelocityoftheliquidmeltlayerasafunctionofmagnetic eldparameter M fordierentPrandtlnumbers Pr whenexternalforce parameter =0.aPr=1,bPr=10,cPr=50,dPr=100........29 4.5Variationofangularvelocityoftheliquidmeltlayerasafunctionofexternalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......31 4.6Variationofangularvelocityoftheliquidmeltlayerasafunctionofmagneticeldparameter M fordierentPrandtlnumbers Pr whenexternal forceparameter =0.aPr=1,bPr=10,cPr=50,dPr=100.....34 4.7Variationofthetemperatureoftheliquidmeltlayerasafunctionofexternalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......36 4.8Variationofthetemperatureoftheliquidmeltlayerasafunctionofmagneticeldparameter M fordierentPrandtlnumbers Ste whenexternal forceparameter =0.aPr=1,bPr=10,cPr=50,dPr=100.....39 x
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4.9Variationofthethicknessoftheliquidmeltlayerasafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......41 4.10Variationofthicknessoftheliquidmeltlayerasafunctionofexternalforce parameter fordierentStefannumbers Ste whenmagneticparameter M =5andPr=10...............................42 4.11Variationofthicknessoftheliquidmeltlayerasafunctionofmagnetic parameter M fordierentStefannumbers Ste whenexternalforceparameter =0.aPr=1,bPr=10,cPr=50,dPr=100...........45 4.12Variationofthemeltingrateasafunctionofexternalforceparameter for dierentStefannumbers Ste whenmagneticparameter M =0.aPr=1, bPr=10,cPr=50,dPr=100.......................48 4.13Variationofthemeltingrateasafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =5and Pr=10.....................................49 4.14Variationofthemeltingrateasafunctionofmagneticparameter M fordierentStefannumbers Ste whenexternalforceparameter =0. aPr=1,bPr=10,cPr=50,dPr=100..................51 4.15Variationoftheheattransferrateevaluatedatheaterwallasafunctionof externalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100.......54 4.16Variationofheattransferratesevaluatedatphasechangelocationasa functionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =0,aPr=1,bPr=10,cPr=50,dPr=10057 4.17Variationofheattransferratesevaluatedatheaterwallasafunctionof magneticparameter M fordierentStefannumbers Ste whenexternal forceparameter =0aPr=1,bPr=10,cPr=50,dPr=100....60 xi
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4.18Variationofheattransferratesevaluatedatphasechangelocationasa functionofmagneticparameter M fordierentStefannumbers Ste when externalforceparameter =0,aPr=1,bPr=10,cPr=50,dPr=10062 4.19Variationoftangentialshearstressevaluatedatwallasafunctionof externalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0,aPr=1,bPr=10,cPr=50,dPr=100.......65 4.20Variationoftangentialshearstressevaluatedatphasechangelocation asafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =0,aPr=1,bPr=10,cPr=50, dPr=100..................................68 4.21Variationoftangentialshearstressevaluatedatwallasafunctionof magneticparameter M fordierentStefannumbers Ste whenexternal forceparameter Ste =0,aPr=1,bPr=10,cPr=50,dPr=100..70 4.22Variationoftangentialshearstressevaluatedatphasechangelocation asafunctionofmagneticparameter M fordierentStefannumbers Ste whenexternalforceparameter Ste =0,aPr=1,bPr=10,cPr=50, dPr=100..................................73 4.23Variationofradialshearstressevaluatedatwallasafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0,aPr=1,bPr=10,cPr=50,dPr=100......75 4.24Variationofradialshearstressevaluatedatphasechangelocationasa functionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =0aPr=1,bPr=10,cPr=50,dPr=10078 4.25Variationofradialshearstressevaluatedatheaterwallasafunctionof magneticelds M fordierentStefannumbers Ste whenexternalforce parameter =0aPr=1,bPr=10,cPr=50,dPr=100........80 xii
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4.26Variationofradialshearstressevaluatedatphasechangelocationasa functionofmagneticelds M fordierentStefannumbers Ste whenexternalforceparameter =0aPr=1,bPr=10,cPr=50,dPr=100.83 xiii
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CHAPTERI INTRODUCTION 1.1Introduction Contactmeltingisaphysicalthermaltransportprocessandaccompaniedbya uidmotioninathinregion,whenthereisaforcedcontactbetweenasolidphase changematerialPCMandaheater.Typically,theheaterismaintainedathigh temperatures.Theheatercanbesubjectedtoconstantheatuxoritismaintained ataconstanttemperature.ThesolidPCMmayvarybasedonthetypeofmelting process,forexampleiceblockformeltingoficerelatedprocessandsolidblockmade ofmetalinmanufacturingindustries.Applicationofcontactmeltingcanbefairly foundinallengineeringindustrieslikemanufacturing,aerospace,shipbuildingetc. 1.2LiteratureReviewonClosecontactmelting Closecontactmeltingisalwayscharacterizedbyaliquidlmformationbetween therubbingsolidsurfaces.Developmentofliquidlmisachievedbyitsownviscous frictionfromitspreviouslayerofmeltingsolid[1].Themodeofheattransferbetween theliquidlmformedisprimarilyrelatedtoheatconduction.Convectionheattransferaredominatedinliquidlmswhichhavelargermagnitudeofthickness.Inmostof thecasesofclosecontactmeltingtheliquidlmthicknesswhichseparatesheaterand PCMsolidaresmall,sincethemeltremovalisachievedwithmultipleprocesses[2]. Contactmeltingprocesscanbewidelygeneralizedasclosecontactmeltingandxed contactmelting.Thereisarelativemotionbetweenthesourceofheatandmelting solidsinclosecontactmelting.Eithertheheatsourcecanbemotionorthemelting substanceisinmotion[3].Closecontactmeltingoersmainlytwoadvantagesover xedcontactmelting,i.etheliquidmeltformationispreventedbecauseoftherelative motionandthemeltisataloweraveragetemperatureinthesystem.Internalclose contactmeltingisdenedwhenthePhaseChangeMaterialmeltsinsidetheheater sourceenclosure,whereastheexternalclosecontactmeltingisdenedwhenthePCM 1
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meltsaroundanembeddedheatsourcebody[4].Ourworkmainlyfocusesonexternal closecontactmeltingcase.Closecontactmeltingapplicationsarespreadacrosswide rangeofengineeringapplications.Someofthetypicalapplicationsinvolveselfburial wastedisposalsofradioactiveelements,reactorcoremeltdowninnuclearindustries [5],applicationsingeophysics,spacecraftandmaterialprocessingindustries[2].Contactmeltingprocessesinsideclosedenclosuresareimportantliquidsolidphasechange processes,whichareusedintechnologyapplicationswherelargeamountsofenergy transfertakeplace.Forexample,airconditioningpumps,latentheatthermalenergy storagesystems,solarwaterheatingconductorsandheatpumps[6]. 1.3LiteratureReviewonCloseContactMeltingwithRotatingDisk Meltremovalformedinbetweentheheaterplateandphasechangematerialis importanttoenhancetheheattransferratebetweenthesolidbodies.Thethinnerthe meltlayerthicknessis,thelargeristheheattransferratethroughheatconduction [7].Therearevariousmethodsofmeltremovalmethods,liketheuseofmagnetic eldsandgravitationalforces[8].Meltremovalprocessusingtheweightofmelting solidwasalsostudiedpreviously[9].Inallthecasesofcontactmeltingthereisa basicphasechangeofmeltingsubstance.Rotatingheaterisemployedinreduced gravityexperiments,wherethemeltremovalisachievedwiththecentrifugalforces generatedbyrotatingheater[10].Materialprocessingapplicationscarriedoutin microgravityexperimentsarealsosomeoftheusefulapplicationsofrotatingdisk contactmelting.Manyauthorshaveanalyzedandstudiedthegoverningequations ofmeltingsolidsunderrotation.Exactnumericalandanalyticalsolutionwerealso developedfordeterminingthemeltingrate[10]. 1.4LiteratureReviewonMagnetohydrodynamics MagnetohydrodynamicsMHDisthestudyoftheinteractionofmagneticeects onthehydrodynamicpropertiesofmovinguids.Therehasbeenextensivestudy 2
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madeonmagneticeectsonowparametersandheattransferrates.Moststudiesin MHDhavebeencarriedouttodeterminethehydrodynamicstabilityduringcertain boundaryconditions[11,12].References[13,14]havestudiedMHDeectsonheat transferratesinrotatingdisksandnaturalconvectionstudiesrespectively.Primary applicationsofMHDowsonrotatingdisksincludehighspeedturbinemaintenance, rotarypartsystemsandcoolingofnuclearreactors.Itstheoreticalapplicationsincludespacephysics,geophysicsandastronomy.InitialideasofowaroundtherotatingdisksintheabsenceofmagneticeldeectswerepioneeredbyVonkarmann in[15].Hedevelopedamodelwithgoverningequationswhichcaptureall oftheessentialphysicsinvolvedinowundergoingrotation.Hisapproachtowards thisproblemwithsimilaritysolutionsresultedintermsofnondimensionalvariables, whichsimpliedthenonlineargoverningequationstoordinarydierentialequations. SuchaapproachcanbeextendedtoMHDowoverrotatingdisks[13].Ingeneralit isfoundthatinvariousthermaltransportprocesses,adecreaseintheuidvelocities andheattransferratesoccurbecauseofthepresenceofmagneticeld[16]. 1.5PresentWork Combinedeectsofmagneticeldsandrotationhavebeenapotentialareaof research.Aspertheknowledgeoftheauthorthereisnopriorstudyintheliterature oncontactmeltingprocesseswhensubjectedtobothrotation,magneticeldsand externalpressurebymeltingsolid.Throughinsightsintosucheectsachievedbya detailedmodelingofthecomplexnonlinearprocessesinvolved,onemaythendevise controltechniquestocontrolthequalitiesofcontactmeltingprocedure.Thisresearch mainlyaddressestheinuenceofmagneticeldondirectcontactmeltingprocessundergoingrotation.Inthisregard,amathematicalmodelinvolvingthreedimensional mass,momentumandenergyequationsincylindricalcoordinatescoupledwithand interfacialphasechangeconditionisdevelopedforthisrotatingsystem.Magnetic eldeectsareaccountedforbymeansoftheadditionalbodyforceterms.Usingap3
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propriatenondimensionalsimilaritytransformation,thegoverningpartialdierential equationsarethentransformedintoasetofsimilarityordinarydierentialequations. Anonlinearnitedierencesolverforthesimilarityequationsrepresentingowand thermaltransportinrotatingsystemandaccompaniedbyphasechangeandmagneticeldeectsaredeveloped.Allthevelocitycomponentsofthemeltlayer,heat transferratebetweentheheatingsurface,meltandmeltingsolid,rateofmelting andthicknessofthemeltlayerarestudiedforvaryingvaluesofmagneticeldsand rotationalspeedsintermsofcharacteristicnondimensionalparameters.Someimportantphysicalparametersofengineeringinterestsliketangentialshearstressand surfaceshearstressintheradialdirectionoftherotatingheaterarealsostudied.This completedissertationisdividedintovechapters.Thischapterprimarilydealswith literaturereviewontheproblem.Chapter2detailsaboutthemathematicalapproach takentowardsstudyingthecontactmeltingduetocombinedeectsofrotationand magneticelds.Itpresentsthegoverningequationsincylindricalcoordinates.All thenondimensionalparametersaredenedinchapter3whichalsodiscussesthesimilaritytransformationsandnumericalsolutionprocedureonnitedierencemethod. Chapter4presentsallkeyresultsanddiscussionsandchapter5providesasummary andconclusionsarisingfromthisthesiswork. 4
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CHAPTERII MATHEMATICALMODELLING 2.1Introduction Thischapterdealswiththedescriptionoftheanalyticalapproachusedtomodel thecontactmeltingproblemsubjectedtobothmagneticeldsandrotation.The physicalproblemconsistsofasolidmeltingblock,consideredasthephasechange materialPCMplacedonrotatingdiskheater.Thewholesetupissubjectedtoan externalmagneticeld.Foreaseofanalysisthephasechangematerialismaintained atitsmeltingtemperatureandtherotatingheaterismaintainedatatemperature higherthanthemeltingtemperatureofthephasechangematerial.Asthecontact meltingprocesscontinues,thephasechangematerialundergoesmeltinganditis separatedfromheaterwithathinlayerofmeltbeingformedinbetweenthem.Melt removalisachievedwithacentrifugalforcegeneratedfromtherotationofheater. Liquidmeltwhichisowingoutwards,hasvariousvelocitycomponentsindierent directionsofthecoordinatesystem.Eectsoftherateofrotationoftheheateralong withtheappliedexternalmagneticeldisalsostudiedonthesevelocitycomponents aswellasthoseonthemeltingrateandthicknessofmeltlayerformationandheat transferrates.ConsistentandsimpliedformsoftheMaxwellsequationsarethen usedtoobtaintheinducedmagneticelds,whichresultsintheLorentzforceinuencingthedynamicsofthethinliquidlmgeneratedbythecontactmeltingprocess. 2.2ComputationalDomainDescription Aschematicpresentationofthecomputationaldomainconsideredisdescribed intheFig.2.1.Ameltingsolid,phasechangematerialisdescribedintheformof parallelepipedcylinder,anditsdimensionsaredenedas L r 0 ,where L istheinitial heightand r 0 isradiusofmeltingsolid.Itismaintainedatitsmeltingtemperature T m .Thisisplacedonacylindricalheaterwhichismaintainedatatemperature T w androtatedwithanangularrotationrate .Thicknessofthemeltlayerinthecontact 5
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meltingprocessisdenedas andthemeltlayerinitiallycoversaradius r 0 andthen owsoutwards.Averticaldownwardforcewhichisweightofthesolidisactingon themeltingsolidwhichalsohelpsintheliquidmeltremovalprocessinadditionto centrifugalforcegeneratedduetorotation.Thepressuredevelopedbecauseofthe weightofthemeltingsolidisconsideredas p 0 andgeneratesthesqueezingeect. Anexternalmagneticeldisappliedtothissetup,whichactsperpendiculartothe directionofliquidmeltremovalasshowninFig2.1.Thestrengthofthismagnetic eldisdenotedas B 0 .Thisexternalmagneticeldgeneratesvariousforcecomponents indierentdirectionswhichwillbediscussedinlaterpartofthischapter. Weconsiderthefollowingassumptionstoproceedfurtherfordevelopingthegoverningequations Flowoftheliquidmeltwhichisdirectedradiallyoutwardislaminar Bylookingatthecomputationaldomainwecanconcludeazimuthalsymmetry. Thisindicateswecanneglectthe @ @ termsingoverningequations. Heaterismaintainedatconstanttemperaturedescribedearlier. Meltingprocessisassumedtobequasisteady. Meltliquidlayergeneratedisconsideredtobeincompressible,viscousandNewtonianuid. 6
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Figure2.1:Computationaldomainforthecontactmeltingproblemundergoing rotationinthepresenceofmagneticeld 2.3MagneticEects Alltheeectsrelatedtothemagnetohydrodynamicsareunderstoodwiththree importantlawsnamelyFaradayslawofinduction,AmpereslawandLorentzsforce. Thecouplingbetweenthecontactmeltingprocessandmagneticeldcanbestudied bythecombinedeectsofuidmechanicsandelectromagneticeects.Theyalso involveanenergytransportphenomenonatthephasechangelocationduetolatent heateectsandalsointheliquidmeltviaconductionandconvection.Relative movementfromtheliquidmeltandappliedexternalmagneticeldinducese.m.f electromotiveforceduetotheFaradayslaw.Underlyingconceptsinducedbythe externalmagneticeldcanbecapturedbytheadditionofaforcetermintothe 7
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governingequationsasmentionedearlier.Thistakestheform F = J B ,where J isthecurrentdensityand B representsthemagneticinductionvector.Theinduced currentdensity J whichcanbedenedviaAmperelawofmagnetohydrodynamicsas J = e E + V B .Here E istheelectriceldstrength, V B istheinduced currentand e istheelectricalconductivity.Previousstudiesandliteratureshave shownthatwecanneglectthe e E termwhichthensimpliesthemagneticforce termreferredtotheLorentzforceas F = e V B .Thisadditionalforcehas aradialcomponentandacircumferentialcomponent,whichmainlyaecttheow parametersofliquidmelt[13].Thus F r = )]TJ/F36 11.9552 Tf 10.693 0 Td [( e V r B o 2 istheradialcomponentand while F = )]TJ/F36 11.9552 Tf 10.694 0 Td [( e V B o 2 representsthecircumferentialcomponentofLorentzforce. Bothofthesetermsareincludedinthegoverningequationstoincludethephysics generatedbyexternalmagneticeldsinthenextsection. 2.4GoverningEquations Basicfundamentalequationswhichgoverntheproblemstatementarethemass continuityequations,momentumcontinuityequationsandenergyequation.Physics attheinterfaceduetothephasechangearisingvialatentheateectscanbecaptured withStefancondition. Tofacilitatemodellingandanalysis,werepresentthegoverningequationsinthe threedimensionalcylindricalcoordinates,inwhichthevelocityeldiswrittenas V = V r ;V ;V z .Firstweconsiderthemasscontinuityequationrepresentedby[10] 1 r @ rv r @r + @v z @z =0.1 ThecomponentsoftheNavierstokesequationsincludingtheLorentzforceeects canthenbewrittenas v r @v r @r + v z @v r @z )]TJ/F19 11.9552 Tf 13.151 9.321 Td [(v 2 r = )]TJ/F15 11.9552 Tf 10.586 8.087 Td [(1 @p @r + @ 2 v r @r 2 + 1 r @v r @r + @ 2 v r @z 2 )]TJ/F19 11.9552 Tf 13.21 8.087 Td [(v r r 2 )]TJ/F19 11.9552 Tf 13.151 8.087 Td [( e v r B 2 0 .2 v r @v @r + v z @v @z + v r v r = @ 2 v @r 2 + 1 r @v @r + @ 2 v @z 2 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(v r 2 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [( e v B 2 0 .3 8
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v r @v z @r + v z @v z @z = )]TJ/F15 11.9552 Tf 10.587 8.087 Td [(1 @p @z + @ 2 v z @r 2 + 1 r @v z @r + @ 2 v z @z 2 .4 Here,Eq..2isthemomentumequationinradialdirection,Eq..3inthe angulardirectionandEq..4intheaxialdirection. v r isthevelocitydenedin radialdirection, v representsthevelocitydenedincircumferentialdirectionand v z isthevelocitydenedinaxialdirection. p = p r;z isthelocalpressureintheliquid meltlayer,whosecumulativeeectbalancestheloadactingonthemeltingsolid P 0 Theenergybalanceequationisthenrepresentedas v r @T @r + v z @T @z = @ 2 T @r 2 + 1 r @T @r + @ 2 T @z 2 .5 Basedonthecomputationaldomain,boundaryconditionsforthisproblemare denedattwodierentlocations.Oneattheheaterwall z =0,andanotheratthe phasechangelocationi.eattheinterface z = .Boundaryconditionswhichare appliedtothetothecomputationaldomainareasfollows. v r =0 v = r!v z =0 T = T w atz =0.6 v r =0 v =0 T = T m atz = .7 )]TJ/F19 11.9552 Tf 9.299 0 Td [(k @T @z j z = = h sf V z z = .8 HereEq..8istheenergybalanceattheinterfacelocation,i.etheStefan condition.Intheaboveequations,theuidproperties k and h sf represent thedensity,kinematicviscosity,thermaldiusivity,thermalconductivityandlatent heatofmeltingrespectively. 9
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CHAPTERIII NONDIMENSIONALPARAMETERS 3.1Introduction GoverningequationsforcontactmeltingproblemundergoingrotationandsubjectedtomagneticeldsarepartialdierentialequationsPDEs.Solvingthose partialdierentialequationsrequiresspecializednumericalmethodsandconsiderablecomputationalresources.Convertingthemintoordinarydierentialequations ODEswouldsimplifythesolutionprocedureandsignicantlyreducerequirements ofcomputationalresources.Nondimensionalsimilarityvariablesareusedinconvertingthepartialdierentialequationstoordinarydierentialequations.Thisprimary approachofimplementingthenondimensionalparameterstotherotatingdiskproblemwasrstdevelopedbyVonKarmaninhisclassicalviscouspumpowproblem analysis.Theproblemisconcernedabouttheowofsemiinniteuidonverylarge atdiskwhichisbeingrotatedataparticularcircularfrequency.OurproblemstatementissimilartoVonKarmanviscouspump.Butourthesisworkdealswithnite liquidlmthicknesswhichisdierentfromsemiinnitesetupinviscouspump.We adoptsimilarapproachtoconvertourpartialdierentialequationstoordinarydifferentialequationswithchangesinboundaryconditions;alsoforanitedomainin theradialdirectionbytakingintoconsiderationthephasechangeandmagneticeld eects. 3.2NonDimensionalVariable Thelistofprimarynondimensionalvariablesusedinouranalysistosimplyand solveourgoverningequationsofthephysicalmodelareasfollows = z 1 = 2 .1 F = V r r! .2 10
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G = V r! .3 H = V z 1 = 2 .4 P = [ p r;z )]TJ/F19 11.9552 Tf 11.955 0 Td [(p r; ] .5 = T w )]TJ/F19 11.9552 Tf 11.956 0 Td [(T T w )]TJ/F19 11.9552 Tf 11.955 0 Td [(T m .6 3.3TransformedGoverningEquations Asdiscussedearlierafterapplyingthesetransformationvariablestothegoverning equationsmentionedintheprevioussection,wecantransformthemtoordinary dierentialequationsODEs.Werstapplythenondimensionalvariablestothe masscontinuityEq..1giving 1 r @ rFr! @r + @ )]TJ/F19 11.9552 Tf 5.48 9.684 Td [(H 1 = 2 @z =0.7 whichafterdierentiatingyields 2 !F + !H 0 1 = 2 != 1 = 2 =0.8 Aftersimplifyingthiswegetfollowingsimilarityequation H 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 F .9 Weapplysimilarkindofapproachwhenweaimtoreplacethemomentumgoverningequationwiththeircorrespondingnondimensionalvariables.Wenowmainly focusourattentiontogoverningEq..2.Aspartofthisderivation,wewillintroduceanondimensionalparameter characterisingexternalloadingandrotation eectandnondimensionalmagneticledparameter M Tobeginderivationthatintroducestheexternalforceparameter ,werstneglectmagnetictermandredenepressuretermintheproblemstatement.Fromthe previousliteratureithasbeenconcludedthatthepressureisindependentofradius 11
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[10].So,thederivativeofpressurewithrespecttoradiusisconstantwhichtakethe formoffollowingequation )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 r @p @r = C: .10 Intergratingthisequationwithlimitsfrom r to r 0 ,whicharethelimitsencompassing theliquidmeltlmyields p r; = C 2 r 2 o )]TJ/F19 11.9552 Tf 11.955 0 Td [(r 2 .11 Aftersolvingfortheconstantbybalancingthepressurein z direction,weget p r; = 2 p 0 r 2 0 r 2 0 )]TJ/F19 11.9552 Tf 11.956 0 Td [(r 2 .12 NowweuseEq..12withEq..5whichisthenondimensionalvariablefor pressuretowrite p r;z as p r;z = P + p r; .13 Now,weapplyEqs..2.4onEq..2toget r! 2 F 2 + r! 2 FH 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(r! 2 G 2 = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @p @r + F! r + r!F 00 )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(F! r .14 Byreplacing p r;z inaboveequationwithEqs..13and.12andrearranging, weget r! 2 F 2 + r! 2 FH 0 )]TJ/F19 11.9552 Tf 10.021 0 Td [(r! 2 G 2 = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @r P + 2 p 0 r 2 0 r 2 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(r 2 + h r!F 00 i .15 Aftereliminatingthecommontermsonbothsides,weobtain F 2 + F 0 H )]TJ/F19 11.9552 Tf 11.955 0 Td [(G 2 = 4 p 0 r 2 0 2 + F 00 .16 Nowwereplacethevariable F with H 0 fromEq..9andrearrangingterms,weget H 000 = HH 00 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( H 0 2 2 +2 G 2 +2 .17 Here, representsthenondimensionalexternalforceparameterdenedby = 4 p 0 r 2 0 2 .18 12
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Now,toderivethenondimensionalmagneticparameter M weemploythesame approachasweusedabove.Butinderivingat M weneglectthegradientofpressure withradiusinEq..2andincludethemagneticterm e v r B 2 0 .Wepartiallyrearrange termsfromEq..2toget v r @v r @r + v z @v r @z )]TJ/F19 11.9552 Tf 13.151 9.321 Td [(v 2 r = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [(@p @r + @ 2 v r @r 2 + 1 r @v r @r + @ 2 v r @z 2 )]TJ/F19 11.9552 Tf 13.21 8.088 Td [(v r r 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [( e v r B 2 0 .19 NowapplyingthenondimensionalvariablesEq..2.4intotheaboveequation, weget h r! 2 F 2 + r! 2 FH 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(r! 2 G 2 i = F! r + r!F 00 )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(F! r )]TJ/F19 11.9552 Tf 11.955 0 Td [( e B 2 0 Fr! .20 Aftersimplicationandeliminatingcommontermsonbothsidesofequation,we obtain F 2 + F 0 H )]TJ/F19 11.9552 Tf 11.955 0 Td [(G 2 = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [( e B 2 0 F + F 00 .21 Wenowidentifyanewmagneticparameter M as M = e B 2 0 .22 So,nallyconsideringthemagneticeldandrotationeects,thetransformedgoverningequationbecome H 000 = HH 00 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [( H 0 2 2 +2 G 2 +2 + MH 0 .23 Wefollowthesamesetofstepsinconvertingtheothergoverningequationsinto transformedsimilarityequations.SofromEq..3 r! 2 FG + H 1 = 2 r!G 0 != 1 = 2 = G! r + r! 2 G 00 )]TJ/F19 11.9552 Tf 13.151 8.088 Td [(G! r )]TJ/F19 11.9552 Tf 13.151 8.088 Td [( e B 2 0 Gr! whichthenbecomes FG + HG 0 + FG = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [( e B 2 0 G + G 00 Aftersimplifyingandrearrangingthenaltransformedgoverningequation,weget G 00 = HG 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(GH 0 + MG .24 13
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Thenconsideringthelastcomponentofthemomentumequation,i.eEq..4 H 1 = 2 1 = 2 H 0 1 = 2 = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @z P + 2 p 0 r 2 0 r 2 0 )]TJ/F19 11.9552 Tf 11.955 0 Td [(r 2 + h 1 = 2 H 00 i andrearrangingitas H 1 = 2 1 = 2 H 0 != 1 = 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(P 0 1 = 2 + 1 = 2 H 00 H! 3 = 2 1 = 2 = )]TJ/F19 11.9552 Tf 9.299 0 Td [(P 0 3 = 2 1 = 2 + H 00 3 = 2 1 = 2 Hence,afterrearrangingthenaltransformedcomponentofthemomentumsimilarityequationbecomes P 0 = H 00 )]TJ/F19 11.9552 Tf 11.955 0 Td [(HH 0 .25 TotransformtheenergyequationEq..5,weshoulddene T fromEq..6.So wehave T = T m )]TJ/F19 11.9552 Tf 11.941 0 Td [(T w + T w .Hence,applyingthetransformationonEq..5,we get H 1 = 2 0 T m )]TJ/F19 11.9552 Tf 11.956 0 Td [(T w 1 = 2 = T m )]TJ/F19 11.9552 Tf 11.955 0 Td [(T w 00 whichthenbecomes H 0 T m )]TJ/F19 11.9552 Tf 11.955 0 Td [(T w = T m )]TJ/F19 11.9552 Tf 11.955 0 Td [(T w 00 Simplifyingthis,wegetthesimilarityenergyequationas 00 = PrH 0 .26 Boundaryconditionsarealsotransformedbasedonthenondimensionalvariable. Theyarelistedbelow H 0 =0 ;G =1 ;H =0 ; =0 at =0.27 14
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H 0 =0 ;G =0 0 =1 ;at = .28 Ste Pr = )]TJ/F19 11.9552 Tf 9.298 0 Td [(H 0 .29 HereEq..29isthesimilarityformofStefancondition,whichintroducestwo additionalnondimensionalparameters,i.eStefannumber Ste andPrandtlnumber Pr denedas Ste = c p T w )]TJ/F19 11.9552 Tf 11.956 0 Td [(T m h sf Pr = .30 where c p isthespecicheatofthemeltlayer Solvingtheaboveequationswithnitedierencemethodasdiscussedinthe nextsectionyieldthevelocityandthermaleldsaswellasthetransportproperties atboundaries.Inturn,suchresultshelpindetermininghowtherotationofheater andmagneticeldsaectthecontactmeltingprocess. 3.4FiniteDierenceMethod Finitedierencemethodisanumericalapproachinsolvingthedierentialequations,bydiscretizingthederivativesusedintheformulation.Theprinciplebehind thisapproachisthederivativesinthedierentialequationsaresubstitutedwithapproximationswhicharederivedbylinearcombinationoffunctionvaluesatthegrid points.Aconcreteexampleofthebasicprincipleofthenitedierencemethodcan beseenasfollows f 0 x = g .31 issubstitutedwith f x + h )]TJ/F19 11.9552 Tf 11.955 0 Td [(f x h = f 0 x = g .32 Thecomputationaldomainisthendividedintomultiplegridpoints.Thisprocess ofdiscretizingthedomainiscalledmeshing.Inordertoachieveaccurateresultsthe 15
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computationaldomainisdividedintomultiplegridpointstoachieveanemesh onthecomputationaldomain.Thesegridpointsareincrementedlinearlyuntilthe completecomputationaldomainiscoveredforanalysis. Thederivativesingoverningequationsintheproblemstatementarereplaced with2ndorderapproximationsatallgridpointsasspeciedbelow.Theseequations areappliedontoallthegridpointsincomputationdomainandrequiredresultsare obtained.Forexample,theequation H 000 = HH 00 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [( H 0 2 2 +2 G 2 +2 + MH 0 H i +2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 H i +1 +2 H i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(H i )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 2 3 = H i H i +1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 H i + H i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( H i +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(H i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 2 +2 G 2 i +2 + M H i +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(H i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 2 .33 andsimilarityequation G 00 = HG 0 )]TJ/F19 11.9552 Tf 11.956 0 Td [(GH 0 + MG becomesafterdiscretizationthefollowing G i +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 G i + G i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 = H i G i +1 )]TJ/F19 11.9552 Tf 11.956 0 Td [(G i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F19 11.9552 Tf 13.151 8.087 Td [(H i +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(H i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 G i + MG i .34 WethenconverttheaboveequationsintoamatrixformtoemploytheTridiagonalmatrixTDMAsolutionmethodanddeterminetheresults.Wecanonly solve2ndorderequationswiththeTridiagonalmatrixmethodand,sotheabove3rd orderequationisreducedtoa2ndorderequation.Inthisregard,weassume H 0 = A sotheequation H 000 )]TJ/F19 11.9552 Tf 11.955 0 Td [(HH 00 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( H 0 2 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(MH 0 =2 G 2 +2 changestoasimplerformgivenby A 00 )]TJ/F19 11.9552 Tf 11.955 0 Td [(H i A 0 + A i A i 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(MA i =2 G 2 i +2 .35 16
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Nowreplacingthederivativeswithapproximations,weget A i +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 A i + A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F19 11.9552 Tf 17.517 8.087 Td [(H i 2 A i +1 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 + A i A i 2 )]TJ/F19 11.9552 Tf 11.956 0 Td [(MA i =2 G 2 i +2 andthengroupingdierenttermsascoecientsof A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 A i and A i +1 as A i +1 1 2 )]TJ/F19 11.9552 Tf 17.516 8.088 Td [(H i 2 + A i )]TJ/F15 11.9552 Tf 22.466 8.088 Td [(2 2 + A i 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(M + A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 1 2 + H i 2 =2 G 2 i +2 .36 Fortheinteriorgridpoints,centraldierncingapproachisemployedandnearthe boundaries,eitheraforwardorabackwarddierencingschemeisusedasappropriate. Similarapproachisusedforvariousnonlinearsimilarityequations,whichsatisfying theboundaryconditions,includingtheStefanconditiontoobtainthenumericalsolutionforasetofcharacteristicparameterssuchas M Pr Ste wheretheunknown nondimnesionalmeltlayerthickness = 1 = 2 isobtainedaspartofthesolution. 17
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CHAPTERIV RESULTSANDDISCUSSION 4.1Introduction Thischapterfocusontheresultsobtainedfromthephysicalmodelconsideredin thepreviouschapter.Allthevariableswhichareofprimaryinteresttothisstudyare plottedandrespectiveconclusionsarederivedfromthem.Theparameterswhichare analysedinthisthesisareasfollows: Liquidlmvelocityeldandtemperaturedistribution Meltlmthickness Meltingrateofsolidmaterialundergoingphasechange Heattransferratesattheheaterwallandalsoatthephasechangeinterface location. Radialshearstressandtangentialshearstress,bothevaluatedatrotatingheater wallandalsoatthephasechangeinterfacelocation. Followingindividualsectionsofthischapterdealwiththeeectofaparticular singleparameterandtherespectivetrendsobtainedgraphicallyarediscussedindetail.Alltheabovequantitiesarestudiedasafunctionofthedimensionlessexternal forceparameter andthedimensionlessstrengthofexternallyappliedmagneticeld M .Inaddition,dierentPrandtlnumbers Pr areconsideredtotakeintoaccount dierentuidswhilechangingrotationrates andmagneticelds M .AlsotheStefanconditionhelpsustointerprettheeectofphasechangephenomenoninmelting process.Hence,theinuenceofphasechangeprocessisincludedintheresultsin theformofchangingStefannumbers Ste .Eachsectionisdividedintotwosubsectionsandeachofthesubsectiondealswithaparticularvariablechangingwith and changingwith M 18
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4.2VelocityandTemperatureproleswithinMeltLayer Whenaliquidmeltlayerisformedduringthemeltingprocess,externalforce andexternalmagneticeldstrength M signicantlyaectitsvelocityandtemperaturedistributions.Studyofthesevelocitycomponentsintheradial,axialandangular directionisimportantintheanalysisofheattransferandmeltingrates.Inthefollowingsubsections,wediscussindetailtheeectof and M ontheradial,axial andangularvelocitycomponentsforvariousvaluesofStefanandPrandtlnumbers. Theseguresareplottedasafunctionofthedimensionlessaxialcoordinate onthe abscissaandadimensionlessvelocitycomponentontheordinate.Itisobservedthat whenchanging or M ,themaximumpossiblevaluesof changeontheabscissain theseguresindicatingthechangeinthethicknessvalueofthemeltlayerforthat particularvalueof and M 4.2.1Radialvelocitycomponent Velocitycomponentoftheliquidmeltlayeralongtheradialdirectionisdened asradialvelocity.Itisconvertedintoanondimensionalvariable F fromEq.3.2 inChapter3. 19
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4.2.1.1Eectofexternalforceonradialvelocityofthemeltlayer a b 20
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c d Figure4.1:Variationofradialvelocityoftheliquidmeltlayerasafunctionof externalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 21
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Figures4.1adshowthevariationof F vs asafunctionof fordierent Pr .Itcanbeseenthattheradialvelocityincreaseswithincreasingexternalload parameter .Withanincreasein ,thecentrifugalforcesactingintheradialdirection ontheliquidmeltlayerincreases,andwecanalsoseeincreasedslopewithhigher valuesoftheradialvelocitiesduetoenhancedsqueezingeectarisingfromincreasing .IncreasingthePrandtlnumberisseentodecreasethevelocitymagnitudesbecause ofincreasedmomentumdiusivitytotheliquidmeltduetoviscouseects. 4.2.1.2Eectofmagneticeldonradialvelocityofthemeltlayer a 22
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b c 23
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d Figure4.2:Variationofradialvelocityoftheliquidmeltlayerasafunctionof magneticeldparameter M fordierentPrandtlnumbers Pr whenexternalforce parameter =0.aPr=1,bPr=10,cPr=50,dPr=100. Bycontrast,asshowninFigs.4.2ad,theeectofmagneticeldrepresented by M isseentoreducethemagnitudeoftheradialvelocitycomponent.Itisevident thatasthemagneticeldstrengthincreasesitdecreasesthevelocityvaluesdueto theretardingeectoftheadditionalforcesarisingfromthemagneticeldeectsi.e Lorentzforces.TheseLorentzforcesactintheoppositedirectionofradialvelocity. Furthermore,itisfoundthatthereisadecreaseinthevelocitymagnitudeswith anincreaseinthePrandtlnumber.Interestingly,forlowerPrandtlnumberuids athighermagneticeldsthereisaexceptionalincreaseinthemaximumvalueof whichhasbroadenedthevelocityprolesduetoarelativelythickermeltlayer,which canbeseeninFig.4.2a. 4.2.2Axialvelocitycomponent Velocityofmeltlayercomponentdenedalongthedirectionofthecentralaxis whichpassesthroughmeltandheaterisdenedastheaxialvelocity.Itisalso 24
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convertedintoanondimensionalvariable H fromEq..3.Thequantity )]TJ/F19 11.9552 Tf 9.298 0 Td [(H in theseguresindicateit'sowdirectionoppositetothecoordinateaxis. 4.2.2.1Eectofexternalforceonaxialvelocityofthemeltlayer a b 25
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c d Figure4.3:Variationofaxialvelocityoftheliquidmeltlayerasafunctionof externalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 26
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Figures.4.3adshowtheeectof ontheaxialvelocityprolefordierent valuesof Pr .Itisseenthatincreasing hasasimilareectontheaxialvelocityprolesbyincreasingthemeltowgeneratedduetohigherexternalloadsandcentrifugal forcesduringcontactmelting.Theslopesof H arefoundtoincreasewithincreasein .Themaximumvaluesof i.emeltlayerthicknessarealsoseentodecreasewith increasein .Also,asbefore,increasingPrandtlnumberdecreasestheaxialvelocity. ForlowerPrandtlnumberuidswithhighervaluesof ,largermeltlayerthicknessis observedwhichisbecauseofgreaterrelativeeectofthermaldiusivityandhigher centrifugalforcesactingonmelt,causingagreatamountofthesolidbeingmelted away. 4.2.2.2Eectofmagneticeldonaxialvelocityofthemeltlayer a 27
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b c 28
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d Figure4.4:Variationofaxialvelocityoftheliquidmeltlayerasafunctionof magneticeldparameter M fordierentPrandtlnumbers Pr whenexternalforce parameter =0.aPr=1,bPr=10,cPr=50,dPr=100. Ontheotherhand,increasingmagneticeldsdecreasestheaxialvelocitymagnitudesbecauseoftheretardingLorentzforcesactingontheliquidmeltlayerseenin Figs.4.4ad.Thisdecreaseinthevelocitymagnitudesareaccompaniedbyan increaseinthemeltlmthickness.Inaddition,suchadecreaseinvelocityisfurther intensiedwithincreaseinthePrandtlnumber. 4.2.3Angularvelocitycomponent Velocitydenedalongthecircumferentialdirectionoftherotatingheaterisdenedasangularvelocitycomponent.Afternondimensionalizationitisrewrittenin termsofthevariable G 29
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4.2.3.1Eectofexternalforceonangularvelocityofthemeltlayer a b 30
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c d Figure4.5:Variationofangularvelocityoftheliquidmeltlayerasafunctionof externalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 31
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Eectofexternalloadparameter ontheangularvelocityprolesfordierent Pr areshowninFigs.4.5ad.Increasing increasesslopeofvelocityprolesof G .Butwhenobservingforaparticularvalueof ,increasein decreasesthevelocity magnitudeswhichisdierentfromothervelocityproles.Thisisasaresultofthe massconservationinthemeltlayer.Tobalancethemeltlayermasscomingintoand goingoutofsystemthereisacorrespondingdecreaseintheangularvelocity.Slope ofthegraphsarealsoseentodecreasewithincreaseinPrandtlnumbers.Higher Prandtlnumberuidsvelocityprolesseemtobestraightlineswithaconstantslope whencomparedwithlowerPrandtlnumberuidcases. 4.2.3.2Eectofmagneticeldsonangularvelocityofthemeltlayer a 32
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b c 33
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d Figure4.6:Variationofangularvelocityoftheliquidmeltlayerasafunctionof magneticeldparameter M fordierentPrandtlnumbers Pr whenexternalforce parameter =0.aPr=1,bPr=10,cPr=50,dPr=100. Increasingthemagneticeldstrengthsisseentodecreasethemagnitudesofthe velocityprolesandalsoincreasetheirslopesseeFigs.4.6ad.Theeect ofmagneticeldisfoundtobemorepronouncedforlowerPrandtlnumberuids ratherthanforhigherPrandtlnumberuids.LookingatFigs.4.6aditisseen thatathigherPrandtlnumber,variationbetweenthevelocityprolesarefoundto beinsignicant. 4.2.4Temperaturedistribution Studyoftemperaturedistributionwithinthemeltformedisimportanttounderstandtheheattransferratesbetweenheater,meltandmeltingsolid.Afternondimensionalizingtemperatureisrepresentedwithnondimensionalvariable 34
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4.2.4.1Eectofexternalforceontemperaturedistributionofmeltlayer a b 35
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c d Figure4.7:Variationofthetemperatureoftheliquidmeltlayerasafunctionof externalforceparameter fordierentPrandtlnumbers Pr whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 36
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AsshowninFigs.4.7ad,temperatureproleswithinthemeltlayershowa constantincreaseintheirslopeswithincreaseintheexternalloadparameter .In addition,thethicknessofthemeltlayeraccompanyingsuchavariationintemperatureisfoundtodecrease.IncreasingthePrandtlnumberalsodecreasesmeltlayer thickness. 4.2.4.2Eectofmagneticeldsontemperaturedistributionofthemelt layer a 37
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b c 38
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d Figure4.8:Variationofthetemperatureoftheliquidmeltlayerasafunctionof magneticeldparameter M fordierentPrandtlnumbers Ste whenexternalforce parameter =0.aPr=1,bPr=10,cPr=50,dPr=100. Increasingthemagneticparameter M decreasestheslopesofthetemperature prolesseeFigs.4.8ad.Highestslopesforthetemperatureisregistered at M =0.AthigherPrandtlnumbers,thereisnotappreciabledierenceinthe temperatureproleswithchangingexternalmagneticelds M 4.3MeltFilmThickness Asdiscussedearlierthemeltingsolidisseparatedfromrotatingheaterwithathin layerofmeltdevelopedinbetweenthem.Soasthemeltingprogressesthethickness oftheliquidmeltincreasesandheaterisseparatedfrommeltingsolidbyathickness whichbecomesaconstantatsteadystate.Weconsidertheexternalforceparameter takingonvaluesof0,0.5,1and5,themagneticparameter M as0,0.5,1,3and 5,theStefannumber Ste as0,0.05,0.1,0.5and1,andPrandtlnumber Pr as1,10, 39
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50,100.Wethenconsiderthevariationofdimensionlessmeltlayerthickness as eitherafunctionof or M bykeepingtheotherparameterasconstant. 4.3.1Eectofexternalforceonmeltthelayerthickness a b 40
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c d Figure4.9:Variationofthethicknessoftheliquidmeltlayerasafunctionof externalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 41
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Figures.4.9adpresentsthevariationofthemeltlayerthickness asa functionoftheexternalforceparameter .Thisgurealsorepresentstheeectof changingtheStefannumberasaparameterforvariousvaluesofPrandtlnumberon themeltlayerthickness.Itisevidentthatthethicknessofthemeltisreducedas increasesforagivenStefannumber,whilebycontrastitisincreasedasStefannumber increasesforagive .AlsohigherPrandtluidsretainthinnerlayers.Themagnitude ofthevariationof oritsslopeishigherforlowervaluesof andforhighervalues oftheStefannumber.AsPrandtlnumberdecreases,asharpdecreaseinmagnitudes ofthethicknessisobservedathighStefannumbersandlower .Increasingexternal forceparameter facilitatesgreaterrateofmeltremoval,whichisformedinbetween therotatingheaterandmeltingsolidandhencethelayerbecomesthinner.And increaseinStefannumberindicatesadditionofmoresensibleheattosystemrelative tolatentheat,whilethedecreaseinPrandtlnumberindicatesthegreaterroleofthe thermaldiusivity,bothofwhichcausethemeltlayertobecomethicker. Figure4.10:Variationofthicknessoftheliquidmeltlayerasafunctionofexternal forceparameter fordierentStefannumbers Ste whenmagneticparameter M =5andPr=10. 42
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Figure4.10isrepresentsthecombinedeectoftheexternalforceparameter andthemagneticparameter M ,whichissettobe M =5.Sowhencomparingwith Fig4.9bwhichisfor M =0nomagneticeldcase,weseeincreasedvalues ofthicknessbecauseoftheadditionalretardingmagneticforcesactingonthemelt, whichinhibitsitsremoval.ItisnoticedthatthetrendofStefannumberissimilar evenunderthecombinedeectof and M butthereissignicantincreaseinthe magnitudeofthicknesswhencomparedtothecasewithoutamagneticeld. 4.3.2Eectofthethemagneticeldonthemeltlayerthickness a 43
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b c 44
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d Figure4.11:Variationofthicknessoftheliquidmeltlayerasafunctionofmagnetic parameter M fordierentStefannumbers Ste whenexternalforceparameter =0. aPr=1,bPr=10,cPr=50,dPr=100. Figures4.11adshoweectofchangingtheexternalmagneticledparameter M onthemeltlmthicknessfordierentsetsof Pr ,bykeeping Ste asaconstant parameterineachcase.Thevariationsin initiatedbyincreasingmagneticeldis dierentfromincreasingtheexternalforce.Itisseenthatthemeltlayerbecomes thickerwithincreaseinthemagneticnumber M ,whichiscontrastingwiththebehaviourobservedwhentheexternalforceparameterisincreased.Andalsoanincrease intheStefannumberandadecreaseinthePrandtlnumberisseentoincreaseslm thicknessformedbetweenheaterplateandmeltingsolid.WhenobservingFig4.11 a,itseenthatforlowPrandtlnumberowsthereisasharpincreaseinmeltlm thickness.Forexample,theincreaseinthicknesswhenthemagneticnumberchanges from3to5isafactorof2foraparticularvalueoftheStefannumberforlower 45
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PrandtlnumberuidswhencomparedtohigherPrandtlnumberuids.So,forlower Prandtlnumberuids,thickermeltlmsaredevelopedwithhighermagneticelds whencoupledwithhigherStefannumbers.Thisincreasingtrendinthicknessfor higher M isbecauseoftheadditionalopposingLorentzforceswhichactonthemelt whensubjectedtoexternalmagneticelds.Thus,higherthemagneticnumber M largerthemagnitudeofretardingmagneticforcesactingonmeltlmthickness. 4.4MeltingRate Meltingrateisdenedastheamountofmeltingsolidgettingconvertedtomelt perunittimeattheinterface.Thiscandefoundbylookingatthemeltvelocity patterninaxialdirectionalongthecentralaxisofrotatingheater.Thismeltvelocity needtoberecordedatthephasechangeinterfacelocation.Hencetheguresin thissectiondepictthenondimensionalmeltingvelocity )]TJ/F19 11.9552 Tf 9.298 0 Td [(H ontheordinateand varying or M ontheabscissa. 4.4.1Eectofexternalforceonmeltingrate a 46
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b c 47
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d Figure4.12:Variationofthemeltingrateasafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =0.aPr=1, bPr=10,cPr=50,dPr=100. Figures4.12adshowtheeectoftheexternalforceparameter andStefan Numberonmeltingrateforvarioussetsof Pr .Aseitherthevalueof orStefan number Ste increases,meltingrateofthesolidblockalsoincreases.Butasthe Prandtlnumberincreases,bycontrastthemeltingratedecreases.Alsoathigher thechangeinthemeltingratewhenStefannumbervariesfrom0.5to1islargerfor Prandtlnumber1.ThesameincreaseinmeltingrateisnotobservedforPrandtl number100inFig.4.12d.Ingeneral,highervaluesof correspondstogreater externalloads,whichenhancesthecontactmeltingprocess.Moreover,whenthe roleofthespecicheatrelativetothelatentheatismorepronouncesasreectedin higher Ste ,itinducesmeltinginthesolidatamorerapidrate.Bycontrast,when themeltlayerhasgreaterrolefortheviscousdiusioneectsi.ehigh Pr itsremoval 48
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isinhibited,whichinturn,slowsdownthemeltgenerationprocess. Figure4.13:Variationofthemeltingrateasafunctionofexternalforceparameter fordierentStefannumbers Ste whenmagneticparameter M =5andPr=10. Thecombinedeectof and M onmeltingrateisshowninFig.4.13.Itisclear that,thereisasmalldecreaseinthemeltingratebecauseofthepresenceofmagnetic eldswhencomparedtothe M =0caseseeFig.4.12b. 4.4.2Eectofmagneticeldonmeltingrate Figures4.14adshowtheeectofmagneticeldsappliedontoliquidmelting ratefordierentsetsfor Pr and Ste .Itisseenthatthemeltratedecreaseswith anincreaseinthemagneticparameter M .Inasimilarfashionasbefore,forhigher valuesoftheStefannumberandlowervaluesofthePrandtlnumber,highmelting ratesareregistered.Moreover,forlowerPrandtlnumberuids,theeectofmagnetic eldismorepronouncedinreducingthemeltingrateforhighvaluesoftheStefan number.WhencomparingFig.4.14aandd,forthesamevalueoftheStefan number,thevariationishigherforlower Pr uidscomparedtohigh Pr uids. 49
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a b 50
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c d Figure4.14:Variationofthemeltingrateasafunctionofmagneticparameter M fordierentStefannumbers Ste whenexternalforceparameter =0.aPr=1, bPr=10,cPr=50,dPr=100. 51
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4.5HeatTransferRate Heattransferrateisanimportantparameterwhendeterminingtheeectof magneticeldsandexternalforceparameterondirectcontactmelting.Variousmodes ofheattransferwhichgenerallyoccurincontactmeltingareheatconductionandheat convection.Heatconductionmodegenerallydominatesincontactmeltingundergoing rotation[2]becauseoftherelativelysmallmeltlmthicknessbetweenthemelting solidandtheheater.Therecanalsobesomeheatconvectionastheliquidmelt layerisformedinbetweenheaterandPCMandissqueezedoutofsystemundergoing rotation.Heattransferratesarestudiedattwodierentlocationsinthisthesis;the rstlocationisdenedattherotatingheaterwallandsecondlocationisdenedatthe phasechangeinterfacelocation.Studyingbothlocationsarecriticalinunderstanding therotationandmagneticeectsonthethermaltransport.Heattransferrateatthe heaterwallprovidesinformationaboutheatexchangebetweenrotatingheaterwall andadjacentliquidmeltformedbesidesheaterwallwhichinturnprovidestheheat inputtoinitiatemeltingontheotherside.Heattransferrateatthephasechange interfacelocationprovidestheheatexchangebetweenliquidmeltandmeltingsolid wherethereisadditionallatentheateecttobeconsidered.Heattransferrateis usuallyisstudiedwithNusseltnumber Nu = hr 0 k .Afterapplyingtransformation Nusseltnumbercanberewrittenintermsofnondimensionaltemperaturegradient )]TJ/F19 11.9552 Tf 9.298 0 Td [( 0 .Hence,theguresthatfollowrepresentvariationbetween )]TJ/F19 11.9552 Tf 9.298 0 Td [( 0 andeither or M .Itmaybenotedthat )]TJ/F19 11.9552 Tf 9.298 0 Td [( 0 indicatestheheattransferrateattheheater wallwhile )]TJ/F19 11.9552 Tf 9.299 0 Td [( 0 representthatatthephasechangelocation. 52
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4.5.1Eectofexternalforceonheattransferrateattheheaterwall a b 53
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c d Figure4.15:Variationoftheheattransferrateevaluatedatheaterwallasa functionofexternalforceparameter fordierentStefannumbers Ste when magneticparameter M =0.aPr=1,bPr=10,cPr=50,dPr=100. 54
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Figures4.15adrepresentstheeectoftheexternalforceparameter ontheheattransferrateattheheaterwallfordierentsetsof Ste and Pr .Clearly increasingtrendintheheattransferratesatwallisobservedwithincreasein aswell asincreasein Pr .Bycontrasttheheattransferratedecreaseswithincreasein Ste Suchalatertrendwith Ste wasnotobservedinourpreviousanalysisofthevariation ofthemeltlmthicknessorthemeltingrate.Increasingtrendofheattransferrate withincreaseinthePrandtlnumbercanbereadilyrelatedtotheconceptsofthe boundarylayertheory.Asthemeltingprocessprogresses,thermalboundarylayer andviscousboundarylayersaredeveloped.Theheattransferrateisrelatedtothe thermalboundarylayerthickness.Basedonascaling,anincreaseinPrandtlnumber indicatesdecreaseinthermalboundarylayerthickness T Pr )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 = 3 [17].Then,the relationbetweenthermalboundarylayerandheattransferratecoecientcanbe inferredfrom h = k T .Hence,wendanincreaseintheheattransferratewith increaseinPrandtlnumber.Theenhancementinthewallheattransferratedueto theexternalforcingparameter canbeexplainedasfollows. As increases,itincreasesthesqueezingowsrateswithinthemeltlayeras noticedinearliersections.Suchhigherowrateswithintheliquidlmwouldthen carryawaygreaterquantitiesofthermalenergyfromtheheaterwall.Ontheother hand,adecreasingin )]TJ/F19 11.9552 Tf 9.298 0 Td [( 0 withStefannumberisnoticed,becausethelargervalue of Ste ,greaterportionofenergyiscontributedtowardsthespecicheateectsinthe solidwithinthesolidratherthanredistributingittotheconvectingheatingwithin theliquidlm. 55
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4.5.2Eectofexternalforceonheattransferrateatthephasechange location a b 56
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c d Figure4.16:Variationofheattransferratesevaluatedatphasechangelocationasa functionofexternalforceparameter fordierentStefannumbers Ste when magneticparameter M =0,aPr=1,bPr=10,cPr=50,dPr=100 57
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Figures4.16adrepresentstheinuenceoftheexternalforceparameter ontheheattransferrateattheinterfacefordierentsetsof Ste and Pr .Itcan beseenthattheheatTransferrateatthephasechangelocationisalsoincreasing withincreaseineither or Pr .Asbefore,itcanbeobservedthatincreasingthe Stefannumberdecreasestheheattransferrateatthephasechangelocation.These trendscanbeexplainedinananalogousmannerasdonefortheheaterwall.Letus nowseethedierencesbetweentheheattransferratepredictionsatthewallandthe interface.Forexample,considerforeachcasewhen Pr =100.Now,whencompared withtheheattransferrateatwalllocationthereisa2%decreaseintheheattransfer rateattheinterfacefor Ste =0 : 05,5%decreasefor Ste =0 : 1,and20%decrease for Ste =0 : 5andalarge32%decreasefor Ste =1.Suchdierencesbetween 0 and 0 with Ste canbeattributedtotheincreasingportionofthethermalenergy beingstoredinthesolidduetospecicheateects. 4.5.3Eectofmagneticeldonheattransferrateattheheaterwall a 58
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b c 59
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d Figure4.17:Variationofheattransferratesevaluatedatheaterwallasafunction ofmagneticparameter M fordierentStefannumbers Ste whenexternalforce parameter =0aPr=1,bPr=10,cPr=50,dPr=100 Figures4.17adrepresenttheeectofmagneticeldontheheattransfer rateattheheaterwall.Itisseenthattheheattransferratedecreaseswithincrease inmagneticelds.Suchadecreaseisseentobemorepronouncedwithincrease inStefannumber.Thereductioninheattransferrateswith M isassociatedwith reducedmeltingratesandlowersqueezelmowarisingduetotheLorentzforces, asdiscussedinearliersections. 60
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4.5.4Eectofmagneticeldsonheattransferratesatthephasechange location a b 61
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c d Figure4.18:Variationofheattransferratesevaluatedatphasechangelocationasa functionofmagneticparameter M fordierentStefannumbers Ste whenexternal forceparameter =0,aPr=1,bPr=10,cPr=50,dPr=100 62
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Asinthepreviouscase,theheattransferrateattheinterface )]TJ/F19 11.9552 Tf 9.299 0 Td [( 0 decreases withanincreaseinthemagneticparameter M seeFigs.4.18ad.Suchan eectisseentobemorepronouncedatlower Pr .Forexample,when Pr =1,there isa43%reductionintheheattransferrateattheinterfaceforthecasewith M =5, whencomparedtothenomagneticeldcase M =0.However,thereduction becomeslessthan4%when Pr =100forsimilarcasesof M .Also,ascomparisons between )]TJ/F19 11.9552 Tf 9.299 0 Td [( 0 and )]TJ/F19 11.9552 Tf 9.298 0 Td [( 0 ,thereisabout34%decreaseforthe )]TJ/F19 11.9552 Tf 9.299 0 Td [( 0 casewhen Ste =1,5%decreasewhen Ste =0 : 1and2.6%reductionfor Ste =0 : 05. 4.6TangentialShearStressandRadialShearStress Wenowdiscusshowthevariouscomponentsoftheshearstressattheheaterwall andtheinterfacelocationvarywithvariouscharacteristicparameter.Theradialshear stressisashearstressactinginradialdirectionontherotatingheater.Ontheother hand,thetangentialshearstressindicatestheamountoftorquerequiredtokeepthe rotatingheaterincircularmotion.Bothcomponentsoftheshearstresscanbederived fromtheNewtonianshearexpression = )]TJ/F19 11.9552 Tf 9.299 0 Td [( @V @z + 1 r @V z @ forthetangentialshear stressand r = )]TJ/F20 7.9701 Tf 6.675 4.976 Td [(@V r @z + @V z @r fortheradialshearstress[13].Weapplyoursimilarity transformationsontheseequationstoconvertthemtoappropriatenondimensional variables.Hencethetangentialshearstressbecomes G 0 attheheaterwalland G 0 attheinterfacelocation.Radialshearstresstaketheform H 00 attheheater walland H 00 atthephasechangelocation.Plotsfortheshearstresscomponents areshownbetween G 0 and G 0 ontheordinatewitheither or M onabscissa; similarly H 00 and H 00 ontheordinateasafunctionofeither or M onabscissa. 63
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4.6.1Tangentialshearstress 4.6.1.1Eectofexternalforceontangentialshearstressattheheater wall a b 64
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c d Figure4.19:Variationoftangentialshearstressevaluatedatwallasafunctionof externalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0,aPr=1,bPr=10,cPr=50,dPr=100 65
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Figures4.19adshowstheeectoftheexternalforceparameter ontangentialshearstress )]TJ/F19 11.9552 Tf 9.299 0 Td [(G 0 attheheaterwallforvarioussetsof Pr and Ste .Itis evidentthatanincreasein resultsinincreaseinthetangentialstress.Bycontrast, foragiven ,thestressappeartodecreaseas Ste increases.Also,anincreasein Pr increasestheheaterwalltangentialshearstress.Nowas increases,whichisan indicationofincreaseintheexternalload,naturallythetorqueactingonrotating heateralsoincreases.Hence,thetangentialshearstress )]TJ/F19 11.9552 Tf 9.298 0 Td [(G 0 requiredtomaintain aconstantcircularmotionfortheheateralsoincreases.IncreaseinthePrandtlnumberanddecreaseintheStefannumbercontributetotheincreasedowoftheliquid meltformedinmelting.Hence,theseparametersalsohelpfacilitatetheincreasein thetangentialshearstress. 4.6.1.2Eectofexternalforceontangentialshearstressatthephase changelocation a 66
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b c 67
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d Figure4.20:Variationoftangentialshearstressevaluatedatphasechangelocation asafunctionofexternalforceparameter fordierentStefannumbers Ste when magneticparameter M =0,aPr=1,bPr=10,cPr=50,dPr=100 AsshowninFigs.4.20ad,thetangentialshearstressattheinterface locationincreaseswithanincreaseintheexternalforceparameter .Similartothat attheheaterwall,thetangentialstressattheinterfacelocationshowssimilartrends withanincreaseinPrandtlnumberandadecreaseintheStefannumber.Itmay benotedthatthedierenceinthemagnitudesoftheshearstressatthewalland interfacelocationarenegligible,as )]TJ/F19 11.9552 Tf 9.299 0 Td [(G 0 isonanaverageonly0.7%smallerwhen comparedto )]TJ/F19 11.9552 Tf 9.298 0 Td [(G 0 68
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4.6.1.3Eectofmagneticeldontangentialshearstressattheheater wall a b 69
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c d Figure4.21:Variationoftangentialshearstressevaluatedatwallasafunctionof magneticparameter M fordierentStefannumbers Ste whenexternalforce parameter Ste =0,aPr=1,bPr=10,cPr=50,dPr=100 70
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Figures4.21adpresentstheinuenceofthemagneticparameter M onthe tangentialstressatthewall )]TJ/F19 11.9552 Tf 9.298 0 Td [(G 0 fordierentsetsof Ste and Pr .Anincreasing trendinthetangentialshearstressisobservedwithincreasingmagneticelds.Itis alsofoundtoincreasewithdecreaseinStefannumberandincreaseinthePrandtl numberlikeintheearliercase.Increaseintangentialstresswithincreaseinmagnetic eldscanbeexplainedwithequation = )]TJ/F19 11.9552 Tf 9.298 0 Td [( @V @z + 1 r @V z @ .Tangentialshearstress increaseswithincreaseinthegradientof V ,whenmagneticeldbecomesstronger, ascanbeseenfromthevelocityprolesdiscussedearlier.Itisalsofoundthatthe shearstresshasamorepronouncedvariationwith M athigher Ste andatlower Pr 4.6.1.4Eectofmagneticeldontangentialshearstressatthephase changelocation a 71
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b c 72
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d Figure4.22:Variationoftangentialshearstressevaluatedatphasechangelocation asafunctionofmagneticparameter M fordierentStefannumbers Ste when externalforceparameter Ste =0,aPr=1,bPr=10,cPr=50,dPr=100 Figures4.22adillustratestheeectofmagneticeldonthetangentialshear stressattheinterfaceforvarioussetsof Ste and Pr .Interestinglythereisachangein tangentialshearstresspatternsatthephasechangelocationasafunctionofmagnetic eld.Thereisadecreasingtrendattheinterface,whencomparedtoincreasingtrend atheaterwalllocation.Eectofthereducedvelocitygradientswith M atthephase changelocationandthemeltowgenerationisanticipatedtobethemaincauseof thischangeintrend.Thischangeintrendneedsfurtherinvestigation,tounderstand thedierentbehaviourofthetangentialstressandtheunderlyingphysicsbehindit. 73
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4.6.2Radialshearstress 4.6.2.1Eectofexternalforceonradialshearstressattheheaterwall a b 74
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c d Figure4.23:Variationofradialshearstressevaluatedatwallasafunctionof externalforceparameter fordierentStefannumbers Ste whenmagnetic parameter M =0,aPr=1,bPr=10,cPr=50,dPr=100 75
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Thevariationofradialshearstressatthewall )]TJ/F19 11.9552 Tf 9.299 0 Td [(H 00 withtheexternalforce parameter fordierentsetsof Ste and Pr arepresentedinFigs4.23ad.It canbenoticedthatthereisasteadyincreaseintheradialshearstressmeasuredat theheaterwallwithincreasein aswellasdueto Ste .As increases,theexternal pressureforceactingonmeltingsolidincreasescausingincreasedradialsqueezingow andaccompaniedbyhigherradialvelocitygradients.Hence,thereincreasingtrend intheradialshearstressisregisteredwith .Athigher ,thevariationsintheradial shearstresswith Ste alsoincreases.Forexample,when Pr =100,at =0,as ste changesfrom0.5to1.0,thevariationin )]TJ/F19 11.9552 Tf 9.299 0 Td [(H 00 isonly0.6%,whileat =5,for similarchangesin Ste ,thevariationin )]TJ/F19 11.9552 Tf 9.299 0 Td [(H 00 takesahighervalueat8%.Alsoit canbeseenasthePrandtlnumberincreases,theradialshearstressatwalldecreases. 4.6.2.2Eectofexternalforceonradialshearstressatthephasechange location a 76
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b c 77
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d Figure4.24:Variationofradialshearstressevaluatedatphasechangelocationasa functionofexternalforceparameter fordierentStefannumbers Ste when magneticparameter M =0aPr=1,bPr=10,cPr=50,dPr=100 Figures4.24abshowthevariationintheradialshearstressevaluatedat thephasechangelocation.Itcanbeseenthatas increasesforagiven Ste or, conversely,as Ste increasesforagiven ,theradialshearstressattheinterface H 00 increases.Bycontrast,theradialshearstressdecreaseswithanincreasein Pr .Theseresultsareverysimilarintrendwiththatanalysedfortheheaterwall intheprevioussection,exceptthatthereisasignchange.Lookingcloselyatradial stressresultsevaluatedattheheaterwall,therespectiveguresareplottedbetween )]TJ/F19 11.9552 Tf 9.299 0 Td [(H 00 and .Butwhenevaluatedatthephasechangelocationgraphsareplotted between H 00 and .Ingeneralradialshearstressischangingfromnegativevalues topositivevalueswhenstudiedovertheentirelengthofthemeltlmthickness. 78
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4.6.2.3Eectofmagneticeldontheradialshearstressattheheater wall a b 79
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c d Figure4.25:Variationofradialshearstressevaluatedatheaterwallasafunctionof magneticelds M fordierentStefannumbers Ste whenexternalforceparameter =0aPr=1,bPr=10,cPr=50,dPr=100 80
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Theeectofmagneticeldparameter M ontheradialshearstressattheheater wall )]TJ/F19 11.9552 Tf 9.298 0 Td [(H 00 fordierentsetsof Ste and Pr isshowninFigs.4.25ad.A decreasingtrendintheradialshearstressisobservedwithincreasein M anddecreasein Ste .Fromtheequation r = )]TJ/F20 7.9701 Tf 6.675 4.976 Td [(@V r @z + @V z @r itisseenthattheradialshear stressincreaseswithincreaseingradientof V r .Increasingthemagneticeldstrength decreasestheslopeofthevelocityproleinradialdirectionasseenfromFigs.4.3 ad.Thisleadstoadecreasingtrendintheradialstress )]TJ/F19 11.9552 Tf 9.299 0 Td [(H 00 asobserved inFigs.4.25ad.InadditiondecreasingradialshearstressseemstobedominantathigherStefannumberespeciallyforlowerPrandtlnumberuids.AsPrandtl numberincreasesvaluesofstressdecreases.ForlowerPrandtlnumberowsdegree ofdecreasingslopeathigherStefannumberismorewhencomparingtolowerStefan numbers.ItisalsoseenthatasPrandtlnumberdecreasesthereisareversalinthe stressvaluetrendasshowninFig.4.26a. 4.6.2.4Eectofmagneticeldonradialshearstressatthephasechange location a 81
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b c 82
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d Figure4.26:Variationofradialshearstressevaluatedatphasechangelocationasa functionofmagneticelds M fordierentStefannumbers Ste whenexternalforce parameter =0aPr=1,bPr=10,cPr=50,dPr=100 Figures4.26adshowtheinuenceof M ontheradialstressattheinterface H 00 fordierentsetsof Ste and Pr .itisevidentthatas M increase,theradial stressattheinterfacedecreases.However,theeectofthe Ste on H 00 exhibitsa complextrendduetotheinteractionofmeltowgenerationandLorentzforceeects. Onanaverage,about34%decreasein H 00 arerecordedwithincreasein M from 0to5forthe Pr =100case. Furthermore,itcanbeseenthatthemagnitudeofradialstressdecreasingwith increaseinPrandtlnumberatthephasechangelocation.Also,asobservedinthe previoussection,forlowerPrandtlnumberuidsadrasticdecreasein H 00 is observedforhigherStefannumbers.Athigher M andlower Pr theradialstress valuesaredecreasingwithincreaseintheStefannumber.Infact,thestressvaluesat 83
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M =5fortheStefannumbers0.5and1aremuchlesserthatfortheStefannumber of0.05and0.1. Whenanalysingtheradialstresswithchanging ,wehavenotedtheradialstress changesitsvaluefromveto+vefromtheheaterwalltophasechangelocation. Similarpatternisobservedwhenstudyingtheradialstressforvariationswithexternal magneticelds. 84
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CHAPTERV SUMMARYANDCONCLUSION Thischapterdealswithconclusionswhicharededucedfromthemythesis.Inthis thesis,aphysicalmodelfortheclosedcontactmeltingprocessundergoingrotation inthepresenceofamagneticeldisdeveloped.Theformulationisbasedonthree dimensionalmass,momentumandenergyequationsalongwiththeStefancondition attheinterfaceandsubjectedtoLorentzforce,whicharewritteninthecylindrical coordinatesystem.Theyarereducedtoasimplersimilarityformulationbymeans ofsimilaritytransformation,whichfacilitatenumericalsolutionbymeansofanite dierencemethod.Thetransportphysicsofthisrotatingcontactmeltingproblem aregovernedbyasetofcharacteristicdimensionlessparameterincludingtheexternal forceparameter ,magneticparameter M ,Stefannumber Ste andPrandtlnumber Pr .Numericalresultsareobtainedtosystematicallystudytheeectsoftheparameteronthevelocityprolesstructureswithintheliquidmeltlayer,meltingrates,heat transferratesandshearstressattheheaterwallandtheinterface.Followingarethe mainndinginthisstudy. Theradialandaxialvelocitycomponentsofthemeltlayerareseentoincrease withincreaseinexternalforcingparameter anddecreasewithPrandtlnumber Pr Onlytheangularvelocitycomponentofmeltlayerisfoundtodecreasewithincreasein tomaintainamassbalanceintherotatingcontactmeltingsystem.Butincreasing magneticeldparameter M decreasesthemagnitudesofthevelocityinalldirections. Inadditionincreaseinthetemperaturevalueswithincreasein isnoticed.Onthe otherhand,theapplicationofthemagneticeldwithincreasingstrengthsisseento causeadecreaseinthemagnitudesofthetemperatureproleswiththemeltlayer. Whenobservingthemeltlmthicknessvariations,thethicknessisfoundtodecreasewithanincreasein .Thisisbecauseas increases,thecentrifugalforce actingonthemeltlmincreasesandhenceitaccommodatesmoremeltremovalfrom 85
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thesystem.AlsoasStefannumberincreases,whichresultsinagreaterportionof theheatsupplygoingtoraisethespecicheatofthemeltingsolid,themeltlm becomesthicker.Greatervariationsinthemeltlayerthicknessforlower andhigher StefannumberisobservedatlowerPrandtlnumberbecauseofmoreprominentrole ofthermaldiusivityeects.Whenstudyingtheeectsofmagneticelds,themelt layerthicknessisseentoincreasewithincreasein M becauseofadditionalretarding Lorentzforcesactingontheliquidmelt.Animportantndingwhichcanbededucedisthatincreasingthemagneticeldstrengthreducesthemagnitudesofmelt velocitiesandheattransferrates.Theseareanalysedinmoredetailfollowingsections. Wenowidentifythat Ste and Pr havesimilareectsonthemeltingratesas theywereonthemeltlayerthickness.Increasing or Ste enhancesthetransferof heatfromtherotatingheaterintothemeltingsolid.Thisisagainmorepronounced forlower Pr uids.Hencedecreasingtrendinthemeltlmthicknessandincreasing increasingvariationsofthemeltingratecanbecorrelatedwitheachotherforrotating contactmeltingsystem. Nextwesummarizethemaindeductionsbasedontheheattransferrateresults. IngeneralitisknownthatincreasingthePrandtlnumberdecreasesthermalboundarylayerthickness.Asthethermalboundarylayerdecreases,itincreaseheattransfer rate.Thushigher Pr uidsgeneratedduringthecontactmeltingprocessisaccompaniedbyhigherheattransferratesatboththeheaterwallandtheinterface.Another conclusionwhichcanbededucedas,atlowervaluesof convectionheattransfer becomessignicantintheliquidmelt.Ontheotherhand,as asincreases,the thicknessofmeltlayerdecreasesandheatconductionbecomesmoredominant.IncreaseintheStefannumberindicatesgreateramountofsensibleheatbeingaddedto system.Hence,thedierenceintheheattransferrateattheheaterwallandphase 86
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changelocationisrelativelylargeofabout32%forhighervaluesofStefannumber. Tangentialshearstressgeneratedduetotheliquidmeltowisfoundtoincrease withincreaseinexternalforceparameter andincreaseinthemagneticparameter M .Boththeseparametersaddadditionaltorquetotherotatingheater,whichresults inanincreaseinshearstress.Buttheyarefoundtodecreasewhenevaluatedatphase changelocationinthecaseofmagneticeldstrengths M Anotherquantityofinterestistheradialshearstress.Itisobservedthatthe radialshearstressincreaseswithincreaseintheexternalforceparameter .Onthe otherhand,theradialshearstressisseentodecreasewithincreasein M .Ingeneral, theseconclusionsareconsistentwiththefactthathighervaluesofexternalforce parameterpromotesgreatermeltremovalrates,whilebycontrasthighermagnetic eldstrengthsisseentodecreasethemagnitudesofmeltlayervelocities. 87
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REFERENCES [1]ABejan.Thefundamentalsofslidingcontactmeltingandfriction. Journalof HeatTransfer ,111:13{20,1989. [2]DGroulxandMLacroix.Studyofclosecontactmeltingoficefromasliding heatedatplate. InternationalJournalofHeatandMassTransfer ,49:4407{ 4416,2006. [3]MKMoallemi,BWWebb,andRViskanta.Anexperimentalandanalytical studyofclosecontactmelting. JournalofHeatTransfer ,108:894{899,1986. [4]HuYaojiang,HSuyi,andSMingheng.Ageneralizedanalysisofclosecontact meltingprocessesintwodimensionalaxisymmetricgeometries. International CommunicationsinHeatandMassTransfer ,26:339{347,1999. [5]MOkaandVPCarey.Auniedtreatmentofthedirectcontactmeltingprocesses inseveralgeometriccases. InternationalCommunicationsinHeatandMass Transfer ,23:187{202,1996. [6]WZChen,QSYang,MQDai,andSMCheng.Ananalyticalsolutionofthe heattransferprocessduringcontactmeltingofphasechangematerialinsidea horizontalellipticaltube. InternationalJournalofEnergyResearch ,22:131{ 140,1998. [7]ASaito,HHong,andOHirokane.Heattransferenhancementinthedirect contactmeltingprocess. InternationalJournalofHeatandMassTransfer 35:295{305,1992. [8]KTaghaviTafreshiandVKDhir.Shapechangeofaninitiallyverticalwall undergoingcondensationdrivenmelting. JournalofHeatTransfer ,105:235{ 240,1983. [9]StevenHEmermanandDLTurcotte.Stokes'sproblemwithmelting. InternationalJournalofHeatandMassTransfer ,26:1625{1630,1983. [10]KTaghavi.Analysisofdirectcontactmeltingunderrotation. JournalofHeat Transfer ,112:137{143,1990. [11]AArikogluandIOzkol.Onthemhdandslipowoverarotatingdiskwithheat transfer. InternationalJournalofNumericalMethodsforHeat&FluidFlow 16:172{184,2006. [12]ODMakinde.Mhdmixedconvectioninteractionwiththermalradiationand nthorderchemicalreactionpastaverticalporousplateembeddedinaporous medium. ChemicalEngineeringCommunications ,198:590{608,2010. 88
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