MEASUREMENTS OF THE THERMAL EXPANSION AND HEAT 
CAPACITY OF METALS BY ELECTROMAGNETIC 
LEVITATION 
 
 
 
Except where reference is made to the work of others, the work described in this thesis is 
my own or was done in collaboration with my advisory committee.  
This thesis does not include proprietary or classified information. 
 
 
 
 
 
 
Baojian Guo 
 
 
 
Certificate of Approval: 
 
 
 
Jeffrey W. Fergus                           Ruel. A. Overfelt, Chair 
Associate Professor                          Professor 
Materials Engineering                        Materials Engineering 
                
 
 
 
Zhongyang Cheng                          Stephen L. McFarland 
Assistant Professor                           Acting Dean 
Materials Engineering                       Graduate School 
MEASUREMENTS OF THE THERMAL EXPANSION AND HEAT 
CAPACITY OF METALS BY ELECTROMAGNETIC 
 LEVITATION 
 
Baojian Guo  
 
A Thesis 
Submitted to 
the Graduate Faculty of 
Auburn University 
in Partial Fulfillment of the 
Requirements for the 
Degree of 
Master of Science 
 
 
Auburn, Alabama 
August 7, 2006
 
iii
MEASUREMENTS OF THE THERMAL EXPANSION AND HEAT 
CAPACITY OF METALS BY ELECTROMAGNETIC 
 LEVITATION 
 
Baojian Guo  
 
Permission is granted to Auburn University to make copies of this thesis at its discretion, 
upon request of individuals or institutions and at their expense. 
The author reserves all publication rights. 
 
 
 
                          Signature of Author 
  
                          Date of Graduation
 
iv
VITA  
         Baojian Guo, son of Liangcai Guo and Feng Xu, was born on January 30, 1977 in 
Ji?ning, P. R. China. He attended Zhejiang University, Hangzhou, China from September 
1996 and obtained a B.S. degree in Mechanical Engineering in 2000. After one year?s 
working for Air Conditioner Division, Midea Co., LTD, Shunde, China, he entered Auburn 
University pursuing his graduate studies in Materials Engineering in January 2002. 
 
v
THESIS ABSTRACT 
 
MEASUREMENTS OF THE THERMAL EXPANSION AND HEAT 
CAPACITY OF METALS BY ELECTROMAGNETIC 
 LEVITATION 
 
Baojian Guo 
Master of Science, August 7, 2006 
(B.S. in M.E., Zhejiang University, Hongzhou, P.R.China, 2000) 
 
92 Typed Pages 
 
Directed by Ruel A. Overfelt 
 
Electromagnetic levitation is a very useful non-contact melting technique that 
can be exploited for measurements of thermophysical properties of many reactive metals 
and alloys. This study focused on thermal expansion and heat capacity measurements 
based on digital image processing and the modulated power method using the 
electromagnetic levitation technique (EML). An improved pixel threshold method was 
developed for accurate determination of the thermal expansion of an axisymmetrically 
levitated droplet.  The modulated power method, originally proposed by Fecht and 
Johnson (1991),  was exploited  for measuring  the   heat capacity of metals in the 
temperature range of around 1300 to 1800 K. Moffat?s uncertainty estimation procedure 
(Moffat, 1998)   was    used   to    theoretically   analyze    the  various   contributions      to               the 
 
vi
experimental uncertainty. A numerical model was developed to examine the sample 
modulated movement and non-uniform temperature distribution effects during the 
modulated heating process. The experimental work used different materials including 
nickel, titanium, zirconium and nickel-based superalloy IN718. The experiments were 
performed with the electromagnetic levitator of Auburn University.  
vii 
ACKNOWLEDGMENTS 
       The author would like to express my grateful appreciation to Dr. Ruel A. 
Overfelt, whose academic guidance, mentorship, support and patience were invaluable 
during the entire course of this research. The experience of studying, working with him 
will have an enormous impact for the career path of the author.    
       The author would like to thank Dr. Deming Wang for his important suggestions 
and help throughout the course of this project.  
       The author also wants to thank my committee members: Dr. Fergus and Dr. 
Cheng for providing unconditional support and valuable suggestions.  
       Sincere appreciation is expressed to my research group members, George 
Teodorescu and Rui Shao, for their support and friendship over the years.  
       Finally, I would like to thank my wife and my parents and other family members 
for their love and understanding. It is their constant support, which makes this thesis 
possible. 
viii 
Style manual or journal used: Auburn University manuals and guides for the preparation 
of theses and dissertations  
Computer Software used: MS Word 2003, MS Excel 2003 and SigmaPlot 8.0  
         
ix 
TABLE OF CONTENTS 
LIST OF TABLES..........................................................................................................xi 
LIST OF FIGURES.......................................................................................................xii 
1.  INTRODUCTION ....................................................................................................1 
2.  LITERATURE REVIEW..........................................................................................5 
2.1. Thermal Expansion Measurement by Image Processing.................................5 
2.2.   Specific Heat Measurement by Modulated Power Method............................7 
3.  EXPERIMENT PROCEDURES AND ANALYTICAL TECHNIQUES..................10 
3.1. Electromagnetic Levitation Apparatus.........................................................10 
3.2. Sample Image Acquisition and Processing...................................................13 
3.3. Modulated Power Method: Control and Data Acquisition............................21 
3.4.   Numerical Model of Modulated Power Method..........................................26 
3.4.1.  Analysis of Sample Movement due to Modulated Power ...................26 
3.4.2.  Analysis of Internal Temperature Field ..............................................28 
4. RESULTS AND DISCUSSION.................................................................................32 
4.1. Sample Movement Effects...........................................................................32 
4.2.   Thermal Expansion of Molten Nickel and Nickel-Based Alloy IN713 ........34 
    4.3.   Modulated Power Specific Heat Measurements..........................................39 
5. CONCLUSIONS.......................................................................................................50 
6. SUGGESTIONS FOR FUTURE RESEARCH ..........................................................52 
x 
REFERENCES..............................................................................................................53 
APPENDICES...............................................................................................................59 
A. THE SOURCE CODE FOR THERMAL EXPANSION MEASUREMENT ..............59 
B.  THE SOURCE CODE FOR NUMERICAL MODEL OF MODULATED 
   POWER METHOD.  ?????????????????????????. 67 
C. THE SOURCE CODE FOR ANALYSIS OF TEMPERATURE DATA .....................76 
xi 
LIST OF TABLES 
Table 3-1.   Parameters of Vulcan-I EML coils . ..........................................................22 
Table 3-2.   Thermophysical properties of liquid nickel samples at the melting  
temperature of 1728 K .............................................................................29 
Table 4-1.   AISI E52100 steel composition (wt %).....................................................34 
Table 4-2.   Nickel-based superalloy IN713 composition (wt %).................................39 
Table 4-3.   Uncertainty Estimates of the Nickel Specific Heat Measurement using   
the EML Modulated Power Method.........................................................47 
Table 4-4.   Uncertainty Estimates of the Titanium Specific Heat Measurement    
using the EML Modulated Power Method................................................48 
Table 4-5.   Uncertainty Estimates of the Zirconium Specific Heat Measurement   
using the EML Modulated Power Method................................................49 
xii 
LIST OF FIGURES 
Figure 1-1.   Schematic sketch of the electromagnetic induction coils with a  
conducting sample in the middle of the system. ..................................... 3 
Figure 2-1.   Radial intensity gradient profile along a levitated droplet. ...................... 8 
Figure 3-1.   Electromagnetic levitator of Auburn University. ..................................... 11 
Figure 3-2.   Functional diagram of experimental setup. .............................................12 
Figure 3-3.   Temperature measurement comparison between pyrometer and  
thermocouple. ........................................................................................14 
Figure 3-4.   A levitated spherical solid nickel sample at 1295 
o
C. ..............................16 
Figure 3-5.   Part of the edge of a droplet comparing a fitted sixth order Legendre 
polynomial with the experimental data. ?????????????   19 
Figure 3-6.   Legendre polynomial of sixth order fit to sample edge  coordinates 
obtained from image processing using the (a) typical gradient method  
and (b) improved threshold method........................................................20 
Figure 3-7.   Vulcan-I EML coil system with a conducting sample in the middle of   
the system . ????????????????????????  22 
Figure 3-8.   Schematic diagram of modulated heating power method.........................24 
Figure 3-9.   Calculated EM force exerted on the levitated sample along axial  
direction (z-axis). ...................................................................................27 
 
xiii 
Figure 3-10.  (a) Spherical coordinates used in the numerical modeling.             
(b) Schematic showing the assumed volumetric heating..........................30 
Figure 4-1.   (a) The peak coil current (lower trace) and resultant sample position 
response (upper trace). (b) Comparison of dynamic temperature  
response of the sample for motionless and moving sample cases.      
?= 0.2 Hz, I
o
=122 A, ?I
o
=3 A, I
m
=10 A..................................................33 
Figure 4-2.   Volume measurements of two steel calibration balls using image 
processing (a) improved threshold method and (b) maximum intensity 
gradient method......................................................................................36 
Figure 4-3.   Density of the steel calibration balls........................................................37 
Figure 4-4.   Experimentally determined density of electromagnetically levitated  
liquid nickel samples..............................................................................38 
Figure 4-5.   Experimentally determined density of electromagnetically levitated 
molten IN713. ........................................................................................41 
Figure 4-6.   Conductive heat loss through the suspension wire and its effect on    
heat capacity calculation. .......................................................................42 
Figure 4-7.   Heat capacity of nickel: present work and data reported in the literature..44 
Figure 4-8.   Heat capacity of titanium: present work and data reported in the  
literature. ??????????????  ...?????????....  45 
Figure 4-9.    Heat capacity of zirconium: present work and data reported in the 
literature.................................................................................................46
1 
1.  INTRODUCTION 
        Information on the temperature dependent thermophysical properties of 
materials is very important in understanding the complex transport phenomena in 
materials processes and in obtaining reliable numerical simulations to optimize 
manufacturing process designs. With the continuous improvements in commercial 
simulation software, the accuracy of the simulation results is often limited by the 
accuracy of the input materials properties. However, such data can be extremely difficult 
to measure if high temperatures are involved and the material exhibits significant 
chemical reactivity with crucibles.  
Electromagnetic levitation (EML) technology has been applied to containerless 
processing of liquid metals from the 1930s with benefits in both process control and 
product quality. Electrically conductive metals can be levitated by magnetic fields under 
clean environmental conditions. The eddy currents provide an effective means for stirring. 
These effects can significantly improve uniformity of compositions and mechanical 
properties. During solidification, heterogeneous nucleation on the container wall is 
eliminated. 
EML  technology has    recently attracted     interest for    thermophysical property 
measurements to alleviate deleterious crucible containment affects in earth-based 
laboratories as well as in space. For example, EML has been utilized in a recent series of 
orbital experiments  with   the  TEMPUS   electromagnetic   levitator  (     Egry  et  al.,     2001; 
2 
Wunderlich et al., 2001). 
Most electromagnetic levitation coils consist of two sets of opposing turns: the 
top coil and the bottom coil. The top coil and bottom coil are wound in different directions. 
When a conductive sample is placed inside of the electromagnetic induction coils that are 
carrying high frequency alternating current, eddy currents are induced in the sample. The 
bottom set of coils always provide the lifting force to oppose gravity due to the mutually 
repulsive interaction between the fields around the coils and the sample?s induced field. 
Since the mutual repulsive nature between the induced field and externally applied 
electromagnetic field, the sample moves towards the weakest part of the applied field. The 
top set of coils provides stability in the lateral direction. A schematic of the applied 
electromagnetic fields inside the electromagnetic levitation coils is shown in Figure 1-1.   
Electromagnetic levitation experiments can also be performed in micro-gravity. 
Space-based electromagnetic levitators (e.g., TEMPUS), designed to operate under 
micro-gravity, use two sets of independent coaxial induction coils. The small positioning 
forces necessary in low-g are provided by one coil while the heating power is provided by 
an additional higher frequency coil. The sample?s temperature can thus be controlled over a 
wide range depending upon the specific sample size and properties.   
Earth-based electromagnetic levitation systems typically use only a single set of 
opposing coils of the quadrapole design and operated at a single frequency to both levitate 
and heat the sample. Although the heating and levitation effects are coupled, such systems  
are easy to fabricate, assemble and operate. An example of such a single coil system is 
t h e     V u l c a n - I         i n s t r u m e n t      o r i g i n a l l y         d e v e l o p e d          f o r         o p e r a t i o n          i n       t h e    e a r t h - b a s e d  
laboratory  as well  as on  parabolic  flights  of NASA?s  KC135  low-g research aircraft  
3 
 
 
 
 
 
Figure 1-1.  Schematic sketch of the electromagnetic induction coils with a conducting 
sample in the middle of the system. 
Top Coil 
Bottom Coil 
Applied 
Magnetic Flux 
4 
(Chen and Overfelt, 1998; Wang et al., 2003). 
A very promising feature of EML is its potential for measuring several important 
thermophysical properties on a single levitated sample over a large temperature range. The 
absence of crucibles eliminates reactions with the melt and high levels of undercooling can 
also be reached. Such applications require sophisticated non-contact diagnostics such as 
two-color pyrometry for temperature measurement and high-speed video analysis for 
characterizing the sample motion. The thermophysical properties of typical interest are 
surface tension, density and thermal expansion, emissivity, specific and latent heats, 
thermal conductivity and electrical conductivity. Reviews of theoretical and experimental 
work in this area are given by Herlach et al. (1993) , Egry et al.(1993) and Bakhtiyarov and 
Overfelt (2002; 2003). 
          More recently, high-temperature electrostatic levitation (HTESL) has been 
developed at NASA?s Jet Propulsion Laboratory (Chung et al., 1996) . HTESL charges a 
sample and then uses electrostatic repulsive forces to levitate and position the charged 
sample.  A separate optical power system (laser, focused lamp, etc.) is needed to heat the 
sample. HTESL systems are inherently unstable and require complex monitoring and 
control systems for reliable operation. Nevertheless, HTESL can be applied to electrical 
non-conducting materials as well as conductive materials.               
         This study intended to develop a new digital image processing method to 
accurately measure the thermal expansion of levitated solid and molten metals and to 
implement the modulated power method of specific heat measurement on different sizes of 
solid samples. In addition, a numerical model was developed to examine the heat transfer 
phenomena and examine the uncertainty in the heat capacity measurements.  
5 
2.  LITERATURE REVIEW 
 
2.1 .  Thermal Expansion Measurement by Image Processing  
Thermal expansion is an important thermophysical property in materials science. 
Since direct physical contact is avoided in EML, standard push rod dilatometry cannot be 
applied for measurements of thermal expansion. Sample images taken with precise optical 
systems are required to characterize the sample sizes and evaluate thermal expansion 
effects. When a sample is levitated and heated by an induction coil, the volumes of 
symmetrical samples can be determined from side view images. However careful coil 
design and fabrication is required to establish stable heating and levitation conditions 
necessary for symmetrical molten droplets in typical earth-based EML laboratories.  
In previous work (El-Mehairy and Ward, 1963), electromagnetically levitated 
samples were photographed from the side using high speed film cameras. A photographic 
enlarger was then used to achieve the necessary resolution for density measurements. The 
edges of the samples for each image were determined by hand, and the volumes were then 
obtained as a body of revolution from the edge profiles. This is a very inefficient method 
and also inevitably involved random errors caused by each individual experimenter.  The 
modern development of CCD (charged-couple device) cameras and digital image 
processing technique enable electromagnetically levitated sample images to be 
automatically recorded  and the recorded images analyzed using digital image processing 
6 
technique. 
The key issue of accurate volume measurement is precise and repeatable 
detection of the droplet edge in the images. An edge is defined as the boundary between 
two regions with relatively distinct gray-level properties. Most edge detection algorithms 
are based on the computation of a local gradient operator where the edge is at the location 
of the maximum intensity gradient. Because of the intrinsic sensitivity of taking derivatives 
from experimental data, the gradient method is very sensitive to noise and has demanding 
requirements for picture quality (Gonzalez and Wintz, 1987; Jain, 1989). Several 
researchers in electromagnetic levitation have adopted first-order derivative gradient 
operators to determine the sample edge from image data (Brillo and Egry, 2003; Chung et 
al., 1996; Damaschke et al., 1998; Gorges et al., 1996; Racz and Egry, 1995).These 
techniques detected the edge by searching for the maximum intensity gradient along radial 
vectors of each picture. Second-order derivative operators have also been reported (Racz 
and Egry, 1995). Figure 2-1 shows a typical intensity gradient profile along a levitated 
sample radial direction. Figure 2-1 (a) is the intensity first-order derivative profile and 
figure 2-1 (b) is the intensity second-order derivative profile. However, as noted above, 
gradient methods have strict requirements on picture quality and thus backlighting 
techniques have been adopted to reduce pixel blooming in CCDs and improve contrast 
(Brillo and Egry, 2003; Chung et al., 1996; Damaschke et al., 1998). Pixel blooming occurs 
when a pixel?s electrical charge exceeds the CCD pixel?s storage limit and then the 
electrical charge overflows to neighboring pixels. 
A robust alternative approach to analysis of noisy images involves threshold 
methods (Gonzalez and Wintz, 1987; Jain, 1989). Racz and Egry (1995) and Gorges et al 
7 
(1996) noticed that hotter and brighter samples exhibited less image contrast than cooler 
samples when the images were due to emitted radiation. These researchers adopted a pixel 
threshold image processing method but still experienced considerable scatter in their 
measured sample volumes. The present research introduces the use of the image of the 
water cooled electromagnetic coil (of known diameter) to determine the specific pixel 
threshold value for edge detection for each individual image. This approach enables 
precise and accurate determination of levitated sample edges without the need for sample 
backlighting. 
2.2.  Specific Heat Measurement by Modulated Power Method  
Accurate knowledge of heat capacity of materials is very important for both 
fundamental studies on phase transformations and optimization of industrial processes. 
Experimental measurements of heat capacity can also be used to derive enthalpy, entropy 
and the Gibbs free energy -- essential thermodynamic parameters.  
 Although differential scanning calorimetry (DSC) is a very fast, accurate, and 
convenient method for numerous applications, serious contamination can occur between 
the sample and crucible when the measurement is carried out at high temperature. 
Non-contact techniques such as electromagnetic levitation (EML) must be applied when 
sample ? crucible interactions may corrupt the experimental data.   
Ohsaka and Schaefers (1992; 1995) successfully coupled the EML technique 
with a drop calorimeter and measured the specific heat of elemental nickel, iron, 
vanadium, and niobium. The enthalpy of each sample was indirectly measured by the 
temperature increase of the calorimeter after dropping the levitated hot sample into the 
calorimeter. This  approach  overcame  the sample ? container  contamination       associated  
8 
Pixel 
0 50 100 150 200 250 300 350
Fi
r
s
t-
or
de
r d
e
r
i
vati
v
e
0
5
10
15
20
25
30
35
 
(a) 
Pixel 
0 50 100 150 200 250 300 350
S
e
c
ond-
or
de
r
 d
e
r
i
v
a
t
i
v
e
-30
-20
-10
0
10
20
30
 
(b)  
Figure  2-1.   Radial intensity gradient profile along a levitated droplet.  
            (a) First-order derivative profile.  (b) Second-order derivative profile. 
9 
with conventional calorimeters. However, it introduced significant experimental 
complexity and only allowed a single point measurement of enthalpy with each sample 
processed. In the pioneering modulated power experiments (Bachmann et al., 1972; 
Sullivan and Seidel, 1968), heat capacity measurement was performed at low 
temperatures with a small sample (1-500mg) and in which a silicon chip was used as the 
sample holder. Fecht and Johnson (1991) and their colleagues (Fecht and Wunderlich, 
1994; Wunderlich et al., 1993; Wunderlich et al., 2000; Wunderlich et al., 2001; 
Wunderlich and Fecht, 1993; Wunderlich and Fecht, 1996; Wunderlich et al., 1997) 
applied the modulated power method to the electromagnetic heating and levitation 
technique. In this application, the electromagnetically heated and levitated sample is 
exposed to a slowly varying sinusoidally-modulated heating power. The temperature 
response of the levitated sample slightly lagged behind the imposed power profile with a 
time constant that depended upon the thermal inertia of the sample. By proper choice of 
the modulation frequency, the transient effects of external and internal thermal 
relaxations can be ignored with errors of only approximately 1% (Fecht and Johnson, 
1991). The unknown specific heat can then be calculated if the sample?s emissive 
properties are known. The electromagnetic levitation technique (Egry et al., 1993; 
Wroughton et al., 1952) combined with the modulation power method is an excellent 
experimental technique that allows containerless heat capacity measurements on 
electrically conductive samples. 
10 
3.  EXPERIMENT PROCEDURES AND ANALYTICAL TECHNIQUES 
3.1 .  Electromagnetic Levitation Apparatus 
A key part of the electromagnetic levitator at Auburn University (Figure 3-1.) is 
the induction coil housed in a vacuum chamber (10
-6
 torr) pumped by a turbomolecular 
pump. A sample handler with a rotational sample selector allows up to 8 samples to be 
processed without opening the vacuum chamber. A commercial 1 kW RF power supply is 
used to provide a high frequency alternating current of approximately 175 amps at 280 kHz 
to the induction coil. The induction coil was configured to impose a quadrupole positioning 
field to keep the sample approximately in the middle of the coil. One of the advantages of 
the quadrapole design is that the system is very simple, easy to make with high degree of 
symmetry and exhibits a stably levitated sample.  
The RF power supply is controlled by a computer and D/A converter using 
RS232 Serial Interfaces. The power supply control signal is composed of a DC component 
and an AC component. A functional diagram of the experimental setup is shown in figure 
3-2. 
11 
 
 
 
 
 
Figure  3-1.  Electromagnetic levitator of Auburn University. 
12 
 
 
 
 
  Figure  3-2.  Functional diagram of experimental setup. 
Computer 
D/A Convert 
Power Supply 
RF Coil 
Pyrometer 
A/D Convert 
Cooling Water 
Camera 
VCR 
13 
A Micron 2-color pyrometer with 0.5% reported uncertainty was used to 
characterize the thermal response of all samples. A solid cylindrical zirconium sample with 
6 mm diameter and 6.2 mm height was suspended by 0.15 mm diameter R-type 
thermocouple wire attached to the sample. Four separate transient heating experiments 
from room temperature to1423 K, 1498 K, 1673 K and 1773 K were conducted in the 
induction coil. The results (See Figure 3-3) indicated that the temperature measurements 
between the pyrometer and the thermocouple agreed to within 0.4% during steady state 
measurements and approximately 0.8% during the transient heating conditions. In 
addition, the pyrometer?s calibration was checked by a comparison with samples of pure 
nickel at the melting temperature of 1728
 
K, the pyrometer showed a slightly smaller 
measurement, the agreement with the reference melting temperature was 0.52%.  
3.2 .  Sample Image Acquisition and Processing  
A CCD camera is used to monitor the experiment and record sample image data 
from the side. Under normal conditions, the sample is at a high enough temperature that the 
emitted radiation is sufficient for direct illumination on the photodetector. All images for 
thermal expansion measurements were taken with a Watec LCL-903HS CCD camera 
(resolution of 768 X 494) at 30 frames per second and a shutter speed of 1/1000 second.  
The central challenge of the image processing is to separate the droplet from the 
background.   The droplet edge is  located at the  transition region of the intensity  profile.  
In the threshold method, if the intensity of a certain pixel is higher than a threshold value, 
the pixel is considered in the region of the droplet. If the intensity of a certain pixel is less 
than a threshold value, the pixel is considered in the background. The key issue is how to 
determine the  threshold  intensity value for each image. The water-cooled  coil tubing (of  
14 
 
 
Time (s)
0 20 40 60 80 100 120 140
T
e
m
p
e
r
a
t
u
re  (
K
)
1300
1400
1500
1600
1700
1800
1900
Pyrometer
Thermocouple
 
Figure  3-3.  Temperature measurement comparison between pyrometer and  
                                            t h e r m o c o u p l e .
15 
known diameter in each image) in the present experiments provides a convenient reference 
to determine (1) the optical system magnification and (2) the unique pixel threshold value 
for each image. See Figure 3-4.   
  The optical system magnification is easily determined using vertical scans 
across the coil tube on images with samples at low temperatures. Typically 30-40 images 
are processed using the maximum intensity gradient method to evaluate the number of 
pixels that represent the tubing diameter. The gradient technique works well with samples 
at low temperatures since blooming effects are negligible. Then the threshold pixel 
intensity value representing the edge of the coil tubing is characterized for every image 
since the number of pixels for the tubing diameter will not change. This process determines 
for each particular image, regardless of the sample brightness, the unique threshold pixel 
value for a bright sample and dark background/foreground.   
 When the threshold transition value of the edge between the bright sample pixels 
and the dark coil pixels is obtained, the center of the sample is identified as follows. A 
matrix of ten horizontal lines and ten vertical lines are scanned across the sample image 
and the sample edges identified using the threshold pixel transition value as evaluated 
above. The center of the sample is then estimated as the average of the ten values with X
0
 = 
(X
edge,max
 - X
edge,min
)/2 and Y
0
 = (Y
edge,max
 - Y
edge,min
)/2. After the sample center is identified, a 
set of 360 equally spaced radial vectors (one degree per vector) is established from the 
sample center outward and the sample edge locations determined using the threshold pixel 
transition value to distinguish the bright sample from the dark background. 
The edge point data are then fit with a sixth order Legendre polynomial. Eq.(1) 
describes the curve fitting function: 
16 
 
 
 
 
 
 
Figure  3-4.  A levitated spherical solid nickel sample at 1295 
o
C. 
                                                      N o t e  t h e  ? h o r i z o n t a l ?  c o i l  b l o c k i n g  p a r t  o f  t h e  i m a g e .  
17 
))(cos()(
6
0
??
?
=
=
j
jjfit
PaR                                (1) 
where P
j
(cos(?)) are the Legendre polynomials and a
j
 are the polynomial coefficients. The 
curve fitting is implemented with a standard least-squares procedure. The curve fitting 
procedure provides a complete definition of the sample?s edge even though the coil blocks 
part of the sample image. In addition, the curve fitting procedure actually improves upon 
the one-pixel resolution of the image. 
Since surface tension ensures a molten sample?s surface remains relatively 
smooth, large deviations in a droplet?s edge data are not physically possible. A typical set 
of edge data and the fitted polynomial are shown in figure 3-5. The data point marked by 
the arrow is clearly an outlier and should not be used in the edge determination procedure. 
The fitting residuals between the locations of the experimental data points and the fitted 
curve are examined and all edge points with a residual greater than 2? are discounted and 
eliminated from the experimental data set. After the outliers are removed, the data are then 
fit again with a new sixth order Legendre polynomial. 
Figure 3-6 shows a comparison of curve fits after the droplet edge data were 
determined by the typical maximum intensity gradient method and the improved threshold 
method as outlined above. The improved threshold method provides much better 
agreement with the experimental data. The improved threshold method was also found to 
be a more accurate approach, as will be discussed later in the experiment results section.    
Assuming that the sample is axisymetric, the droplet volume can then be 
calculated as a body of revolution from the smooth curve as 
18 
                   
?
=
?
???
?
0
)(
3
)(
3
2
dSinRV                                   (2) 
Once the volume is determined, the density is easily calculated if the sample 
mass is known. In the present experiments, the mass of the each sample was carefully 
evaluated before and after the experiment and evaporation was assumed to occur linearly 
with time while the sample was molten. The value of dV/dT for molten metals is of the 
order of 10
-4
 g cm
-3
 K
-1
. The error analysis of Racz and Egry (Racz and Egry, 1995) show 
that edge location using pixel interpolation combined with Legendre polynomial fitting 
enable theoretical volume uncertainties of ?V/V ~ 10
-4
.  
19 
 
 
 
Radius (Pixels)
0 50 100 150 200 250
R
a
d
i
u
s
( p
i
xels)
0
50
100
150
200
250
 
 
Figure 3-5.  Part of the edge of a droplet comparing a fitted sixth order Legendre  
   polynomial with the experimental data. A clearly erroneous  
 experimental data point is identified. 
 
20 
(a)  
      
(b)  
 
Figure  3-6.  Legendre polynomial of sixth order fit to sample edge coordinates,      
             obtained from image processing using the (a) typical gradient method       
               and (b) improved threshold method.
21 
3.3.  Modulated Power Method: Control and Data Acquisition 
Okress et al.(1952), Fromm and Jehn (1965) and Xuan ( 2000) assumed that the 
sample is much smaller than the size of the induction coils and treated the levitated sample 
as a circular current loop. Xuan derived the time-averaged axial (i.e., z) levitation force 
F(z) and power absorption P(z) expressions, for Vulcan-I RF coil system in figure 3-7, as:  
                        
npeak
SxGIzF )(
8
9
)(
2
?=                                  (3) 
and                      
)()(
4
3
)(
21
zHxFIRzP
peak
?
= ??                             (4) 
where     
                     
)(sin4)(sinh4
)2sin((3)2sinh(3
1)(
22
xxxx
xx
xG
+
?
?=                          (5)               
??
?+
?
?+
=
n
nn
nn
n
nn
n
n
zzb
zzb
zzb
b
S
5.222
2
5.122
2
])([
)(
])([
                  (6)   
                
)2cos()2cosh(
)2cos()2cosh()2sin()2sinh(
)(
xx
xxxxxx
xF
?
+?+
=                (7)              
      
2
5.122
2
}
])([
{)(
?
?+
=
n
nn
n
zzb
b
zH                            (8)                
and                           ??? fRx  =                                       (9) 
Here, ? is magnetic permeability of the sample, R is sample?s radius, ? is the   
sample?s electrical conductivity, I
peak
 is the maximum current of the induction coil, and f 
is the current frequency. The geometric configuration for the coil set is summarized in 
Table 3-1. 
22 
 
 
         
 
 
Z1
Z2
Z3
Z4
Z5
Z6
b1
Z axis
 
Figure 3-7.  Vulcan-I EML coil system with a conducting sample in 
     t h e  m i d d l e  o f  t h e  s y s t e m .  
 
Table 3-1.  Parameters of Vulcan-I EML coils. 
  
Coil Set 
radius(mm)x turns 
Z1 
(mm) 
Z2 
(mm) 
Z3 
(mm) 
Z4 
(mm) 
Z5 
(mm) 
Z6 
(mm) 
4.5 x  6 -8.44 -6.24 -4.04 -1.84 1.84 4.04 
           (Origin is located at the middle of the gap between upper coils and lower coils)   
23 
         Figure 3-8 shows a typical thermal response for the model system. In this 
application of the modulation power technique, a spherical sample is heated by a total 
power which can be expressed as: 
         )cos( tPPPP
oototal
?
?
?+?+=                              (10) 
where P
o
 is the steady power, ?P
?
 is the modulation component of power and ?P
o
 is the net 
increase in steady power due to the modulation. The sample?s temperature response also 
exhibits three components: the bias temperature T
o
, related to P
o
, an oscillatory component 
?T
?
 induced by ?P
?,
 and a net increase in bias temperature ?T
o
 due to ?P
o
. Theoretically, 
the amplitude of ?T
?
 is given by (Bachmann et al., 1972; Fecht and Johnson, 1991; 
Sullivan and Seidel, 1968): 
                 
2/12
2
2
1
])()(1[
??
++
?
=? ????
??
?
?
p
VC
P
T                    (11)        
Here ?
1
 is the sample?s external relaxation time and ?
2
 is the sample?s internal relaxation 
time, defined as: 
                 
3
1
4
oSB
p
TA
VC
??
?
? =                                         (12) 
R
VC
p
??
?
?
3
2
4
3
=                                          (13) 
? is sample?s thermal conductivity, ? is sample?s density, V is the sample volume, 
A is the sample surface area and 
SB
?  is Stefan?s constant. As noted by Fecht and Johnson 
(1991), if the modulation frequency is appropriately chosen (~0.1 ? 0.5 Hz for typical 
metal samples in earth-based levitation systems), the transient effects of external and 
internal  thermal  relaxations   can    be    ignored    with  errors  of  only     approximately   1%. 
24 
 
 
 
 
Time (s)
0 10203040506070
T
e
mp
eratu
r
e (
 K)
1860
1880
1900
1920
1940
1960
1980
2000
5
10
15
20
25
o
T?
?T
?
 
T
o
 
P
o
 
? P
?
 
o
P?
A
b
so
rb
e
d
 H
eati
n
g
 
P
o
w
e
r
 (W
)
Sample temperature
Absorbed heating power
 
            Figure  3-8.  Schematic diagram of modulated heating power method 
25 
Thus Eq.(11) reduces to: 
                
p
VC
P
T
??
?
?
?
=?                                    (14) 
A control voltage is applied to the power supply of the RF system, and the current in the 
electromagnetic coil can be represented as: 
                        )cos()( tIItI
mopeak
?+=                            (15) 
Substituting Eq.(15) in Eq.(4), the total power can be formulated as: 
             )2cos()cos(
2
tPtPPPP
oototal
??
??
?+?+?+=                   (16) 
where,                    )()(
4
3
21
zHxFIRP
oo
?
= ??                             (17) 
                      )()(
8
3
21
zHxFIRP
mo
?
=? ??                             (18) 
           )()(
2
3
1
zHxFIIRP
mo
?
=? ??
?
                           (19) 
and                           
o
PP ?=?
?2
                                       (20) 
At steady state in vacuum, the input power P
o
 is just balanced by the radiative 
heat losses. When the coil current is modulated, the bulk sample temperature will rise by 
?T
o
 due to the increase ?P
o
 in average absorbed power as shown in figure 3-8. Although 
the temperature signal should theoretically contain a 2? frequency component, this 
component is <1% due to I
m
<<I
o
. Determination of the specific heat from Eq. (14) requires 
knowledge of the power modulation amplitude ?P
? 
from the sample?s temperature 
response. For a motionless sample, ?P
? 
can be estimated from P
o
 as: 
o
o
m
P
I
I
P
2
=?
?
                                      (21) 
26 
where P
o
 is experimentally obtained using the Stefan-Boltzmann law: 
               ][
4
4
envoSBo
TTAP ?= ??                                    (22)  
in which T
env 
is surrounding environment temperature.  
3.4.  Numerical Model of Modulated Power Method  
3.4.1.  Analysis of Sample Movement due to Modulated Power 
In the presence of modulation current in the form of Eq.(15), the total 
levitation force can be expressed by Eq.(23) when neglecting the 2? frequency 
component. 
     )cos( tFFFF
oototal
?
?
?+?+=                       (23) 
where: 
 
noo
SxGIF )(
8
9
2
?=                              (24)  
 
nmo
SxGIF )(
16
9
2
?=?                            (25) 
nmo
SxGIIF )(
4
9
?
?
=?                           (26)             
With knowledge of the time-averaged levitation force and modulated current 
form, mathematical expressions can be quickly developed for analyzing the sample?s 
oscillatory motion. Figure 3-9 shows a typical theoretical levitation profile on a nickel 
sample in the coil design used in the current experiments. The equilibrium levitation 
position is indicated by the arrow. When the sample deviates from its equilibrium position, 
a restoring force proportional to the displacement is exerted on the sample. Therefore, a 
simple spring mass system can be used to describe the oscillatory motion of levitated 
samples.  
27 
 
 
 
Position of the sample (mm)
-20 -10 0 10 20
Le
v
i
t
a
t
i
on For
c
e
 / (
V
olum
e
 x
 g )
 
      
      
    (
g
/c
c
)
-20
-15
-10
-5
0
5
10
15
20
Lower turns
Upper turns
Net levitation force
EML coil range
 
Figure 3-9.   Calculated EM force exerted on the levitated sample along axial   
        direction (z-axis). Coil Current (peak): 130A, Frequency: 200 KHz.    
     The equilibrium position corresponding to the sample density is indicated
28 
          F
o
 is the steady levitation force in the absence of any power modulation. ?F
o 
is 
the increased bias of the steady levitation force due to the modulation effects. Thus the 
equilibrium position is increased slightly due to the power modulation. ?F
?
 is the 
amplitude of the modulation force term, which will cause the oscillatory motion. Assuming 
that ?F
o
 and ?F
?
 are constant and calculating S
n
 at the new sample equilibrium position 
enables the modulation movement to be fully described by a spring-mass system with an 
external oscillatory driving force. The sample will vibrate in forced harmonic motion 
around its new equilibrium position at the driving frequency ?. The motion of the sample is 
governed by Newton?s second law: 
        ()tFkz
dt
zd
m ?
?
cos
2
2
?=+                               (27) 
where m is the sample mass and k is the spring constant determined from the levitation 
force profile. The natural frequency of typical metal samples in typical levitation systems 
is about 10 Hz. As noted earlier, the modulation frequencies for heat capacity 
measurements are typically 0.1-0.5 Hz, fortunately well below the natural frequency of the 
levitation system.  
3.4.2.  Analysis of Internal Temperature Field                                                    
A simplified heat transfer model was developed considering internal conduction 
and external radiation for spherical solid samples undergoing uniform induction heating 
around an equatorial band extending ?10
o
 (north and south) from the equator. The 
governing heat transfer equation in spherical coordinates (Figure 3-10 (a)) is given by: 
.
2
2
222
2
2
)(
sin
1
)(sin
)(sin
)( Q
T
r
T
rr
T
r
rr
k
t
T
VC
p
+
?
?
?
?
?
?
++=
??
?
???
?
?
???
?
?
?
?
?
?
?
?       (28) 
29 
where the heat generation term 
.
Q only existed in the inductively heated region. The 
penetration depth of induction heating is given by: 
0
2
???
? =                                (29) 
Figure 3-10 (b) is the schematic showing the simplified heating assumed. Table 3-2 lists the 
thermophysical properties of liquid nickel used for the heat transfer analysis. The sample 
size investigated was 4 mm diameter.  
Table 3-2. Thermophysical properties of liquid nickel samples at the melting 
temperature of 1728 K (Brandes and Brook, 1992) .  
 Parameters Value Unit 
density   (?) 
emissivity  (?) 
specific heat  (C
p
) 
thermal conductivity  (k) 
electrical conductivity (?) 
saturated pressure (P
s
) 
7.905 
0.22 
620 
76 
5?10
6 
0.0029 
g cm
-3 
 
J kg
-1
k
-1 
wm
-1
k
-1 
(?m)
-1 
torr 
30 
 
 
(a)  
 
 
 
(b)  
 
Figure  3-10.  (a) Spherical coordinates used in the numerical modeling.  
(b) Schematic showing the assumed volumetric heating. 
31 
The electromagnetic heating power was assumed to be uniformly distributed in a 
shell volume region defined by the sample surface (latitudes from 10? north to 10? south) 
and the penetration depth. Electromagnetic levitation systems typically operate at 100-500 
kHz.  Thus for this analysis of the modulated power method using frequencies of 0.1-0.5 
Hz, the induced heating currents can be assumed to instantaneously rise to their peak 
values. The radiation heat losses were linearized as: 
     )(
1 envlrarad
TTAhq ?=
+
                               (30) 
with                   ))((
22
envlenvlSBra
TTTTh ++= ??                             (31) 
where, the subscripts l, and l+1 denote the times t and t+?t.
32 
4.  RESULTS AND DISCUSSION 
4.1 .  Sample Movement Effects  
The dynamic response of a levitated 4 mm diameter nickel sample in the 
modulated heating process was theoretically calculated as described in Section 3.4.1. A 
typical example of representative result for normal levitation conditions is shown in figure 
4-1.(a). The lower trace of figure 4-1.(a) shows the induction coil peak current calculated 
from Eq.(15) for a modulation frequency ? = 0.2 Hz. The predicted sample motion is 
shown in the upper trace of figure 4-1.(a). As expected, the equilibrium levitation position 
increased after application of the modulation power. In addition, the sample is predicted to 
oscillate about the equilibrium position in phase and at the same frequency as the power 
modulation frequency.   
The theoretical thermal response of the upper pole of the spherical sample to the 
modulated input power is shown in figure 4-1.(b). The thermal response of the upper pole 
also displays the same frequency as the modulation power. Peak temperatures were 
predicted to occur at the equator as expected. In addition, the sample exhibited slightly 
smaller temperature amplitudes for the motionless condition as shown in the figure 4-1.(b), 
and the difference between the motionless and moving sample cases is about 2.1%. 
33 
Time (s)
0 102030405060
P
o
si
t
i
on (
m
m
)
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
C
u
rre
n
t
 (A
)
110
120
130
140
150
Position
Current 
Time (s)
0 102030405060
Tem
p
er
at
u
r
e (
K
)
1890
1895
1900
1905
1910
1915
1920
Modulation movement considered
No modulation movement assumption
(a)
(b)
      
Figure 4-1.   (a) The peak coil current (lower trace) and resultant sample position   
response (upper trace). (b) Comparison of dynamic temperature response 
of the sample for motionless and moving sample cases. ?= 0.2 Hz, I
o
=122 
A, ?I
o
=3 A, I
m
=10 A.  
34 
4.2.  Thermal Expansion of Molten Nickel and Nickel-Based Alloy IN713 
In order to test the accuracy of the present image processing technique, two 
different diameter steel calibration balls (AISI E52100) were electromagnetically heated, 
and the images were recorded from the video camera and analyzed for their volumes. The 
composition of AISI E52100 steel is summarized in Table 4-1.  
Table 4-1.  AISI E52100 steel composition  (wt %). 
 
Element C Cr Fe Mn P S Si 
E52100 0.98-1.1 1.45 97 0.35 Max 0.025 Max 0.025 0.23 
 
The normalized volumes (value divided by the average value) are shown in 
figure 4-2. Figure 4-2(a) shows the volume measurements using the improved threshold 
method, and figure 4-2 (b) shows the volume measurements using the maximum intensity 
gradient method. The result shows that the improved threshold method exhibits a smaller 
standard deviation in terms of volume determination. The difference in these two methods 
arises mainly from the blooming effect, which generated a noisy contour for the heated, 
bright calibration balls. Figure 4-3 shows the density of the calibration balls based on the 
volume determination using the two imaging processing techniques. Compared with the 
literature reference (Esser and Eusterbrock, 1941), the result shows that the improved 
threshold method agrees well with the reference values. The maximum intensity gradient 
method reveals a tendency to overestimate the sample?s volume and lead to a smaller 
density value in agreement with blooming effects. 
The density of a molten nickel sample (99.99% purity) was evaluated from a set 
of image data using the two image analysis techniques. These data were then compared to 
35 
density data from Chung et al.(1996). In that previous work, a very stable electrostatic 
levitator was used to eliminate image degradation from sample translation effects and 
backlighting of the sample eliminated noisy image data from CCD blooming effects. The 
data of Chung et al. exhibited minimum scatter and is believed to accurately represent the 
density of molten nickel. Figure 4-4 shows data of the density of molten nickel from the 
present investigation using the maximum intensity gradient method and the improved 
threshold method as well as the literature (Chung et al., 1996). The result for the liquid 
phase density can be expressed by 
Maximum intensity gradient method: 
           ? (g/cc) = 9.5296 - 0.0010*T  (K)                            (31) 
Improved threshold method: 
          ? (g/cc) = 8.7726-0.0005*T  (K)                             (32) 
Reference (Chung et al., 1996): 
           ? (g/cc) = 9.0391-0.0007*T  (K)                            (33) 
The maximum intensity gradient method does not provide as good an agreement 
with the prior work (Chung et al., 1996) as does the threshold method for direct images 
from bright, glowing samples, presumably due to CCD blooming effects. CCD blooming 
biases the maximum intensity gradient method towards larger estimates of sample sizes 
and consequently lower densities. Although CCD blooming effects can be eliminated by 
backlighting the sample, backlighting does introduce additional instrument complexity.  
The pixel threshold method essentially solves this problem by ?calibrating? the blooming 
effect through measurement of the known coil diameter and establishing the correct 
threshold pixel value for a bright  edge against a dark background. Note that the deviation 
36 
Temperature (K)
1440 1480 1520 1560 1600 1640 1680
V /
 V
a
0.85
0.90
0.95
1.00
1.05
1.10
1.15
3 / 16 inch 
7 /32  inch
 
(a)  
Temperature (K)
1440 1480 1520 1560 1600 1640 1680
V /
 V
a
0.85
0.90
0.95
1.00
1.05
1.10
1.15
3 /16 inch
7 / 32 inch
 
(b)  
Figure 4-2.  Volume measurements of two steel calibration balls using image processing 
(a) improved threshold method and (b) maximum intensity gradient method. 
37 
 
 
Temperature (K)
1350 1400 1450 1500 1550 1600 1650 1700
De
ns
it
y
 (
 g/
c
c
)
6.6
6.8
7.0
7.2
7.4
7.6
7.8
8.0
8.2
Esser et al., 1941
Improved threshold method
Maximum intensity gradient method
?=?0.0002
?=?0.0003
?=?0.0007
 
 
Figure  4-3.  Density of the steel calibration balls. 
38 
 
 
 
 
Temperature (K)
1750 1800 1850 1900 1950 2000
Den
s
i
t
y
 (g
/
c
c)
7.0
7.2
7.4
7.6
7.8
8.0
8.2
Maximum Intensity Gradient
Improved Threshold Method
Chung et al., 1996
    
Figure  4-4.  Experimentally determined density of electromagnetically 
      levitated liquid nickel sample. 
39 
of the maximum gradient method increases as the temperature of the nickel sample 
increases, presumably due to increased CCD blooming from the brighter samples. 
Commercial nickel-based superalloy IN713 is widely used in demanding 
applications due to the alloy?s excellent high temperature strength. The composition of this 
alloy is shown in Table 4-2.  
Table 4-2. Nickel-based superalloy IN713 composition  (wt %). 
 
Element C Cr Mo Fe Ti Al Co Nb 
IN713 0.13 13.89 4.0 0.2 0.9 6.0 0.15 2.0 
  
Unfortunately electromagnetic levitation of such alloys is difficult because the 
electrical conductivities are low and the densities are high. Successful levitation and 
melting of the alloy was achieved after several different coil designs were tested. Volumes 
of the samples were estimated using the image threshold technique. The sample masses 
were measured before and after the experiments and the sample mass during the 
measurements estimated assuming constant evaporation rates while each sample was 
molten. The measured density of IN713 in liquid state with 95% prediction interval of the 
regression line is shown in Figure 4-5. 
4.3.  Modulated Power Specific Heat Measurements 
          Modulation heating and cooling experiments were performed with solid nickel, 
titanium and zirconium samples in the temperature range of 1300 K to 1800 K. The 
samples were suspended in the center of the induction coil with a very small 0.15 mm 
diameter Pt (87%)-Rh (13%) wire.  
A simple conduction model of the heat losses through the wire was developed by 
40 
assuming that one end of the Pt-Rh wire is held at the sample temperature and the other end 
of the wire is held at room temperature. The results are shown in figure 4-6 and suggests 
that the heat losses through the wire varied from about 1.5% of the radiation losses at 1273 
K to around 0.5% of the radiation losses at 1773
 
K for a 6 mm diameter nickel sample.  
        The thermal response of the suspended samples were evaluated after being 
heated to four steady state temperatures and then subjected to a sinusoidally-modulated 
heating power. The total hemispherical emissivity of the samples were determined from 
the spectral normal emissivity measurement by coupling a FTIR spectrometer with the 
electromagnetic levitator (Teodorescu et al., 2005). The sample?s specific heat was then 
calculated from the measurements of T
o
 and ?T
?
 using the methodology described above, 
i.e., Eqs. (14, 21, and 22). The measurements were performed for each material using two 
samples with different diameters as noted in the figures. The reproducibility of 
measurements for individual samples was ?3%.  
The heat capacity measurement of nickel is shown in Fig.4-7. There is  
general agreement between the present results and the data of Booker et al.(1961), 
Cezairliyan and Muelle (1983), Desai (1987), and Krauss and Warncke (1955). Figure 
4-8 shows that the experimentally determined heat capacity of titanium from the present 
work agrees with Maglic et al.(2001), Kaschnitz et al. (2001)and Paradis et al. (2000) up 
to approximately 1600 K. The present data indicate lower heat capacity values than the 
literature values in the 1600-1700 K temperature range. Additional testing is required to 
better understand the reason for this difference. Figure 4-9 shows that the experimentally 
determined heat capacity of zirconium from the present work is in good agreement with 
Hultgren  et al (1973)  and  the  pulse  heating  experiments       reported by  Cezairliyan  et 
41 
 
 
 
Temperature (K)
1650 1660 1670 1680 1690 1700
D
e
n
s
it
y (
g
/cc
)
6.8
6.9
7.0
7.1
7.2
R
2
=0.623
 
     Figure  4-5.  Experimentally determined density of electromagnetically  
 levitated molten IN713.
42 
 
 
 
 
 
Temperature (K)
1400 1600 1800 2000
C
ondu
c
t
i
v
e
 he
a
t
 l
o
s
s
 /
 R
a
di
a
t
i
v
e
 he
a
t
 
l
o
s
s
 (%)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
S
p
ecif
i
c h
eat
 m
easr
eu
m
e
n
t
 e
rro
r (
%
)
0.0
0.1
0.2
0.3
0.4
0.5
Conductive heat loss 
Cp measurement error
 
Figure  4-6.  Conductive heat loss through the suspension wire and its effect on     
             heat capacity calculation.
43 
al.(1974). Alcock et al. (1976) reported slightly higher values of the heat capacity as well 
as greater temperature dependence.   
The least-squares fit polynomial functions that represent the results for heat 
capacity for nickel, titanium and zirconium in the measured temperature range are: 
    Nickel: (1380 ? T ? 1678) 
           
3-82-4
102.923-101.050T0.109-61.885 TTC
p
????+?=        (34) 
Titanium: (1361 ? T ? 1721) 
3-82-4
10194.2101.112-T0.19075.767 TTC
p
??+???+=        (35) 
Zirconium: (1359 ? T ?1686)  
    
3-82-4
10341.3101.442T0.201-121.315 TTC
p
?????+?=        (36) 
where C
p 
is in J ? mol 
-1
 ? K
-1
, and T is in K. In the computation of heat capacity, the 
atomic weights of nickel, titanium and zirconium were taken as 58.693, 47.880 and 
91.224, respectively.  
Moffat?s uncertainty estimation procedure (Moffat, 1998) was used to 
theoretically analyze the various contributions to the experimental uncertainty.  The 
results of these calculations are shown in Table 4-3, 4-4, and 4-5. The total estimated 
uncertainty (95% confidence limits) is approximately 4.5 % for a typical specific heat 
measurement.     
The largest contributor to the uncertainty in specific heat measurement was the 
uncertainty in emissivity (primarily due to the ?0.5% uncertainty in temperature 
measurement of the noncontact pyrometer). Improvements in the accuracy of the 
temperature characterization are possible on measurements of solid samples of pure 
elements evaluated incrementally just below and just above the melting temperature. 
44 
 
 
 
Temperature (K)
1300 1400 1500 1600 1700
C
p
 (
J
/m
o
l
 K
)
24
27
30
33
36
39
42
6.0 mm ( Present results)
5.5 mm (Present results)
Desai,1987
Cezairliyan et al.,1983, 
Krauss et al.,1955
Booker et al.,1961
 
Figure  4-7.  Heat capacity of nickel: present work and data reported in the literature. 
45 
 
 
 
 
Temperature (K)
1400 1500 1600 1700 1800
Cp
 (
J
/m
o
l
 
K
)
22
24
26
28
30
32
34
36
38
40
5.5 mm (Present results)
5.9 mm (Present results)
Kaschnitz  et al.,2001
Boboridis et al.,2002
Paradis et al.,2000
Maglic et al.,2001
 
Figure  4-8.  Heat capacity of titanium: present work and data reported in the literature.
46 
 
 
 
 
Temperature (K)
1300 1400 1500 1600 1700 1800
S
p
e
c
if
i
c
 H
e
a
t
 (
J
/
m
o
l
 K
)
26
28
30
32
34
36
38
40
5.5mm (Present results)
6.0mm  (Present results)
Cezairliy et al.,1974
Alcock et al.,1976
Hultgren et al.,1973
 
 
Figure  4-9.  Heat capacity of zirconium: present work and data reported in the 
literature. 
47 
 
 
 
Table 4-3. Uncertainty Estimates of the Nickel Specific Heat Measurement using the 
EML Modulated Power Method. 
 
 
Parameter             Estimated ?2?         Specific heat change        Specific heat change 
                    confidence limits (%)        (J/mol K)              squared (J/mol K)
2
 
                           
Temperature T=1500(K)           0.5                0.72                       0.51 
Modulation frequency ?=0.2(Hz)   0.2               0.07                      0.005 
Sample mass m=945 (mg)          0.1                0.04                      0.001 
Temperature amplitude ?T
m
=5(K)    0.5               0.18                      0.032 
DC control voltage V
o
= 80 (V)        0.2               0.07                      0.005 
AC control voltage V
m
=21.8(V)      0.2               0.03                      0.001 
Emissivity ?=0.2                 4.0               1.43                       2.05 
Total uncertainty in specific heat, J/mol K  
      
2/12
])([
i
???
                                                       1.61 
  Total % uncertainty in specific heat 
        (35.80 J/mol K)                                                     ?4.5% 
 
48 
 
 
Table 4-4. Uncertainty Estimates of the Titanium Specific Heat Measurement using the 
EML Modulated Power Method. 
 
 
Parameter               Estimated ?2?         Specific heat change     Specific heat change 
                    confidence limits (%)        (J/mol K)             squared (J/mol K)
2
 
                           
Temperature T=1500(K)             0.5               0.648                    0.420 
Modulation frequency ?=0.2(Hz)    0.2               0.065                   0.004 
Sample mass m=710 (mg)           0.1               0.033                   0.001 
Temperature amplitude ?T
m
=8 (K)    0.5               0.162                   0.026 
DC control voltage V
o
=98(V)        0.2                0.065                   0.004 
AC control voltage V
m
=12 (V)       0.2                0.035                   0.001 
Emissivity ?=0.28                 4.0                1.294                   1.674 
Total uncertainty in specific heat, J/mol K  
      
2/12
])([
i
???
                                                      1.46 
  Total % uncertainty in specific heat 
        (32.31J/mol K)                                                    ?4.52% 
 
 
49 
 
 
 
Table 4-5. Uncertainty Estimates of the Zirconium Specific Heat Measurement using the 
EML Modulated Power Method. 
 
 
Parameter               Estimated ?2?         Specific heat change     Specific heat change 
                    confidence limits (%)        (J/mol K)             squared (J/mol K)
2
 
                           
Temperature T=1500(K)             0.5               0.561                   0.315 
Modulation frequency ?=0.2(Hz)     0.2               0.056                   0.003 
Sample mass m=484 (mg)           0.1               0.028                    0.001 
Temperature amplitude ?T
m
=10(K)     0.5                0.140                   0.020 
DC control voltage V
o
=90(V)         0.2               0.056                   0.003 
AC control voltage V
m
=21.8(V)       0.2               0.039                   0.002 
Emissivity ?=0.27                  4.0               1.121                   1.256 
Total uncertainty in specific heat, J/mol K  
      
2/12
])([
i
???
                                                       1.27 
  Total % uncertainty in specific heat 
        (30.29 J/mol K)                                                    ? 4.21% 
 
 
50 
5.  CONCLUSIONS 
This work includes two general parts: (i) thermal expansion measurements and 
(ii) heat capacity measurements in the earth-based electromagnetic levitator of Auburn 
University.  
For thermal expansion measurements, CCD blooming effects from the emitted 
radiation of electromagnetically levitated samples can cause noisy image data and 
decreased precision in measurements of sample size using the traditional maximum 
intensity gradient method. However, the presence in the image of the water cooled copper 
coil presents a convenient image of known size to establish the threshold pixel intensity 
value for the edge for each picture and enables effective calibration of the image for the 
blooming effect. These image-specific pixel threshold values can then be used to determine 
the molten droplet edges. Sixth order Legendre polynomials can then be fit to the droplet 
edge data to obtain accurate and precise measurements of the sample volume assuming that 
axisymetric symmetry prevails. The method was confirmed with measurements on 
precision steel calibration balls and pure nickel and then applied to measurements of the 
density of IN713.  
The modulation power method combined with electromagnetic levitation high 
temperatures, especially when interactions with the crucible and contamination are 
concerns. The largest contributor to the uncertainty in specific heat is the uncertainty in 
total hemispherical emissivity  values used  to  calculate the radiative  power losses. The 
51 
method was successfully applied to measure the heat capacity of pure solid samples of 
nickel, titanium and zirconium suspended on a thin Pt-Ph wire. Although the coupled 
heating power and positioning force for the traditional single coil electromagnetic levitator 
design introduces cyclic translations of the sample, these effects are predicted by a 
numerical model developed in this study to have negligible affect on the measurements. 
52 
6. SUGGESTIONS FOR FUTURE RESEARCH 
         To improve accuracy of the density and heat capacity measurements, the current 
experimental system can be modified in the following ways: 
       The current optical system and temperature measurement system performance can 
be significantly improved by reducing the electromagnetic noise generated by the RF 
power supply. 
        Placing the current CCD camera with a higher resolution CCD camera will 
improve the resolution of image sizes.  
        The modulated power method may also be applied to monitoring the phase 
transformation and measuring the alloy?s latent heat. Application of the modulated power 
method within the mushy zone of an alloy will cause periodic melting and freezing 
processes and induce additional lag in the thermal response of the sample. Measurements 
of an alloy?s specific heat over the range of temperatures associated with the phase 
transformation will produce heat capacity measurements above the baseline expected for a 
single phase. When this apparent heat capacity is plotted versus temperature, a peak will 
naturally result and indicate the phase transformation. Integration of the apparent heat 
capacity curve above the single phase baseline will yield the transformation enthalpy.  
53 
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Thermophysical property measurements using electromagnetic levitation", 
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liquid metals by noncontact techniques", International Journal of Thermophysics, 
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metals: application to copper", Transactions of the American Institute of Mining, 
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Esser, H. and Eusterbrock, H., "Investigation of the thermal expansion of some metals and 
alloys with an improved dilatometer", Archiv fuer das Eisenhuettenwesen, vol.14, 
pp.341-355, 1941 
Fecht, H.-J. and Johnson, W. L., "A conceptual approach for noncontact calorimetry in 
space", Review of  Scientific Instruments, vol.62(5), pp.1299-1303, 1991 
Fecht, H.-J. and Wunderlich, R. K., "Development of containerless modulation calorimetry 
for specific heat measurements of undercooled melts", Materials Science & 
Engineering, A: Structural Materials: Properties, Microstructure and Processing, 
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Gonzalez, R. C. and Wintz, P. A., "Digital Image Processing", pp.331-391, 1987 
Gorges, E.; Racz, L. M.; Schillings, A. and Ergy, I., "Density measurements on levitated 
liquid metal droplets", International Journal of Thermophysics, vol.17 (5), 
pp.1163-1172, 1996 
56 
Herlach, D. M.; Cochrane, R. F.; Egry, I.; Fecht, H. J. and Greer, A. L., "Containerless 
processing in the study of metallic melts and their solidification", International 
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1900 K measured by a millisecond pulse-heating technique", Journal of Thermal 
Analysis and Calorimetry, vol.64 (1), pp.351-356, 2001 
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Zeitschrift fuer Metallkunde, vol.46, pp.61-69, 1955 
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300 and 1900 K", International Journal of Thermophysics, vol.22 (6), 
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undercooled liquids", 4
th
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microgravity materials science research, pp.1-6, 1992 
Okress, E. C.; Wroughton, D. M.; Comentz, G.; Brace, P. H. and R.Kelly, J. C., 
"Electromagnetic Levitation of Solid and Molten Metals", Journal of Applied 
Physics, vol.23, pp.545 - 552, 1952 
Paradis, P.-F. and Rhim, W.-K., "Non-contact measurements of thermophysical properties 
57 
of titanium at high temperature", Journal of Chemical Thermodynamics, vol.32 (1), 
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Racz, L. M. and Egry, I., "Advances in the measurement of density and thermal expansion 
of undercooled liquid metals", Review of Scientific Instruments, vol.66 (8), 
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Schaefers, K.; R?sner-Kuhn, M. and Frohberg, M. G., "Enthalpy measurements  of 
undercooled metals by levitation calorimetry: the pure metals nickel, iron, 
vanadium and niobium", Materials Science and Engineering, vol. A 197, pp.83-90, 
1995 
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Electromagnetic Heated Ni at High Temperature", Materials Science & Technology 
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electromagnetic levitation melting technique", AFS Transactions, vol.111, 
pp.813-824, 2003 
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technique for eliminating crucibles in heating and melting metals", Journal of the 
Electrochemical Society, vol.99, pp.205-211, 1952 
Wunderlich, R. K.; Diefenbach, A.; Willnecker, R. and Fecht, H. J., "Principles of 
non-contact a.c. calorimetry", Containerless Process.: Tech. Appl., Proc. Int. Symp. 
Exp. Methods Mater. Sci. Res., 5th, pp.51-56, 1993 
58 
Wunderlich, R. K.; Ettl, C. and Fecht, H.-J., "Non-contact electromagnetic calorimetry of 
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2000 
Wunderlich, R. K.; Ettl, C. and Fecht, H. J., "Non-contact electromagnetic calorimetry of 
metallic glass forming alloys in reduced gravity", European Space Agency, 
[Special Publication] SP, vol. SP-454 (Vol. 1, First International Symposium on 
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2000, Volume 2), pp.537-544, 2001 
Wunderlich, R. K.; Ettl, C. and Fecht, H. J., "Specific heat and thermal transport 
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1993 
Wunderlich, R. K. and Fecht, H. J., "Measurements of thermophysical properties by 
contactless modulation calorimetry", International Journal of Thermophysics, 
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levitation", M.S. thesis, Auburn University, pp.12-27, 2000
59 
APPENDIX A  
THE SOURCE CODE FOR THERMAL EXPANSION MEASUREMENT  
60 
% Main program is designed coil measurement 
% function  xycenter:  for center detection 
% function  intens: for characteristic intensity detection 
clc; 
clear; 
% function   coil:  for coil detection 
% function   radius:  for edge detection 
 SS=31;   % The first Picture 
 MM=40;   % The last Picture 
 for NN=SS:MM, 
   cd('U:\In713 April 26'); 
   s1=char('p'); 
   s11=char('coil'); 
   s2=int2str(NN); 
   s3=char('.jpg'); 
   name=strcat(s1,s2,s3)      % Image file: pX.jpg 
   datafile=strcat(s11,s2);   % Data  file: cedgeX 
   I=imread(name); 
   cd('H:\Program'); 
   A=rgb2gray(I); 
   corner1=[65,220]; 
   corner2=[1092,1020]; 
   xleft=corner1(1); 
   yupp=corner1(2); 
   xright=corner2(1); 
   ydownp=corner2(2); 
   A=A(yupp:ydownp,xleft:xright); 
   B= im2double(A); 
   [m,n]=size(B); 
   % Find the characteristic intensity 
   % Coil Region 
    mark1=100; 
    mark2=150; 
    mark3=210; 
% Center Position determination 
[centerx,centery]=xycenter(B,mark1,mark3); 
[cyupc,cydownc,coillength,transion]= 
    coil(B,centerx,mark1,mark2,mark3); 
    cd('H:\Program\Data\Coil'); 
    save(datafile, 'cyupc','cydownc','centerx','centery','coillength'); 
   clc; 
   clear; 
end 
% Sub_program  
function [centerx,centery]= xycenter(B,mark1,mark3) 
61 
[m,n]=size(B); 
exline=B(mark1,:); 
[record,index]=max(exline); 
exline1=[index,1]; 
exline2=[index,n]; 
eyline=[mark1,mark1]; 
[cx1,cy1,c1]=improfile(B,exline1,eyline,'bicubic'); 
[cx2,cy2,c2]=improfile(B,exline2,eyline,'bicubic'); 
recordl=0; 
critical=0.46; 
intensity=record-critical*(record-recordl); 
hh=1; 
   while c1(hh)>intensity, 
           hh=hh+1; 
   end           
x1=cx1(hh); 
in1=c1(hh);   % Left edge point 
     hh=1; 
     while c2(hh)>intensity, % 
        hh=hh+1; 
     end 
     x2=cx2(hh); 
     in2=c2(hh);   %%% Right edge point 
     xcenter=(x1+x2)/2; 
     centerx=round(xcenter); 
     xcenter=centerx; 
     % Center y determination 
     eyline=B(:,xcenter); 
     [record,index]=max(eyline); 
     intensity=record-critical*(record-recordl); 
     exline=[xcenter,xcenter]; 
     eyline1=[mark1,1]; 
     eyline2=[mark3,m]; 
     hh=1; 
     [cx1,cy1,c1]=improfile(B,exline,eyline1,'bicubic'); 
     [cx2,cy2,c2]=improfile(B,exline,eyline2,'bicubic'); 
      while c1(hh)>intensity, 
         hh=hh+1; 
      end 
     y1=cy1(hh);  % Upper edge point 
     in1=c1(hh); 
     hh=1; 
% For some reason: the mist in the image, we can only use maximum intensity gradient to 
find the ycenter position 
   recordl=min(c2); 
62 
   if recordl<intensity, 
      while c2(hh)>intensity, 
         hh=hh+1; 
      end 
      y2=cy2(hh); % Lower edge point 
     in2=c2(hh); 
   else 
       dif=-diff(c2); 
     [record,index]=max(dif); 
     y2=cy2(index); 
     in2=c2(index); 
   end 
     
     ycenter=(y1+y2)/2; 
     ycenter=round(ycenter); 
     centery=ycenter; 
%----------------------------------------------- 
clc; 
clear; 
% This program is designed for the correction factor of every group images and the highest 
intensity of the image 
SS=11;       % The start Picture 
MM=30;       % The final Picture 
ww=188.0346; % coil distance 
for NN=SS:MM, 
  cd('H:\Program\Sample image\In718 April06(2)'); 
  s1=char('p'); 
  s11=char('coren'); 
  s2=int2str(NN); 
  s3=char('.jpg'); 
  name=strcat(s1,s2,s3)      % Image File 
  datafile=strcat(s11,s2);   %  Data File 
  I=imread(name); 
   cd('H:\Program'); 
   A=rgb2gray(I); 
   corner1=[65,220]; 
   corner2=[1092,1020]; 
   xleft=corner1(1); 
   yupp=corner1(2); 
   xright=corner2(1); 
   ydownp=corner2(2); 
   A=A(yupp:ydownp,xleft:xright); 
   B= im2double(A); 
   [m,n]=size(B); 
63 
   % Coil Region 
    mark1=480; 
    mark2=600; 
    mark3=705; 
[centerx,centery]=xycenter(B,mark1,mark3); 
  recordh=intens(B,centerx); 
  recordl=0; 
   critical=0.6; 
    ww=188.0346; 
    % Center Line Improfile 
  xline=[centerx,centerx]; 
  yline1=[mark1,mark2]; 
  yline2=[mark3,mark2]; 
  [cxc1,cyc1,cc1]=improfile(B,xline,yline1,'bicubic'); 
  [cxc2,cyc2,cc2]=improfile(B,xline,yline2,'bicubic'); 
  [recordup,indexhup]=max(cc1); 
  [recorddown,indexdown]=max(cc1); 
  recordl=0; 
  ccoil=0; 
while abs(ccoil-ww)>2 & critical>=0.20, 
  intensityup=recordh-critical*(recordh-recordl); 
  intensitydown=recordh-critical*(recordh-recordl); 
   % Searching the upper and lower level 
  indexup=1; 
  while cc1(indexup)>intensityup, 
          indexup=indexup+1; 
  end  
  indexdown=1; 
  while cc2(indexdown)>intensitydown, 
          indexdown=indexdown+1; 
  end   
  cyupc=cyc1(indexup);        % Up edge of coil 
  cydownc=cyc2(indexdown);    % Low edge of coil 
  ccoil=cydownc-cyupc;        %Coil distance 
  critical=critical-0.01; 
end 
cd('H:\Program\Data\Core'); 
save(datafile, 'critical','ccoil','recordh'); 
      clc; 
      clear; 
end 
function intensity=intens(B,centerx) 
  for h=1:10, 
     exline=B(:,centerx); 
     inten(h)=max(exline); 
64 
  end 
  intensity=mean(inten); 
clear; 
clc; 
% read critical value function 
SS=21;       % The start Picture 
MM=25;       % The final Picture 
h=1; 
for NN=SS:MM, 
  cd('H:\Program\Data\Core'); 
  s11=char('coren'); 
  s2=int2str(NN); 
  datafile=strcat(s11,s2);   % Data File 
  load(datafile); 
  value(h)=critical; 
  data(h)=ccoil; 
  intensity(h)=recordh; 
  edgein(h)=intensity(h)*(1-value(h)); 
  h=h+1; 
end 
end 
%  
% This program is designed for edge detection 
% function    xycenter:  for center detection 
% function    intens  : for characteristic intensity detection 
clc; 
clear; 
%% function       coil    :  for coil detection 
%% function       radius  :  for edge detection 
%---------------------------------------------------------- 
 SS=21;      % The first Picture 
 MM=25;      % The last Picture 
for NN=SS:MM, 
   critical=0.31; 
   s1=char('p'); 
   s11=char('cedgenew'); 
   s2=int2str(NN); 
   s3=char('.jpg'); 
   name=strcat(s1,s2,s3)  % Image file: pX.jpg 
   datafile=strcat(s11,s2); % Data  file: cedgeX 
   cd('H:\Program\Sample image\In718 April06(2)'); 
   I=imread(name); 
   cd('H:\Program'); 
   A=rgb2gray(I); 
65 
   corner1=[65,220]; 
   corner2=[1092,1020]; 
   xleft=corner1(1); 
   yupp=corner1(2); 
   xright=corner2(1); 
   ydownp=corner2(2); 
   A=A(yupp:ydownp,xleft:xright); 
   B= im2double(A); 
   [m,n]=size(B); 
 % Find the characteristic intensity 
   % Coil Region 
    mark1=480; 
    mark2=600; 
    mark3=705; 
    % Center Position determination 
    [centerx,centery]=xycenter(B,mark1,mark3); 
    % Edge intensity 
    intensity=intens(B,centerx); 
    recordl=0; 
% intensity=intensity-critical*(intensity-recordl); 
% critical value method 
    intensity=0.6738; 
    % edge detection region 
[cyupc,cydownc,coillength,transion]=coil(B,centerx,mark1,mark2,mark3); 
    % Set the start point 
     if centery<=transion(1), 
      ycenter=450; 
      %ycenter=centery; 
     else 
      ycenter=centery; 
   end  
[ang11,r11]=radius11(B,centerx,centery,ycenter,intensity,transion(1),transion(2)); 
[ang22,r22]=radius22(B,centerx,centery,ycenter,intensity,transion(1),transion(2));  
[ang33,r33]=radius33(B,centerx,centery,ycenter,intensity,transion(1),transion(2)); 
[ang44,r44]=radius44(B,centerx,centery,ycenter,intensity,transion(1),transion(2)); 
ang=[ang11,ang22,ang33,ang44]; 
   r=[r11,r22,r33,r44]; 
   cd('H:\Program\Data\Cedge'); 
   save(datafile, 
'ang','r','ang11','ang22','ang33','ang44','r11','r22','r33','r44','centerx','centery',... 
        'ycenter','intensity','coillength'); 
  clc; 
 clear; 
end 
% Curve fitting 
66 
close all; 
clc; 
clear; 
coil=189.5724; 
cd('H:\Program\Data\Cedge\In713'); 
load cedgen30; 
%Fitting and showing 
cd('H:\Program'); 
[fity1,y1,reangright,p1]=right(r11,r44,ang11,ang44); 
[fity2,y2,reangleft,p2]=left(r22,r33,ang22,ang33); 
show(p1,p2); 
hold on; 
polar(ang,r,'x'); 
hold off; 
% Radius Filter 
[filterr1,filterang1,filterr4,filterang4]=filterright(r11,ang11,r44,ang44,fity1); 
[filterr2,filterang2,filterr3,filterang3]=filterleft(r22,ang22,r33,ang33,fity2); 
% Fitting and showing 
[fityy1,yy1,reangrighty,p1y]=right(filterr1,filterr4,filterang1,filterang4); 
[fityy2,yy2,reanglefty,p2y]=left(filterr2,filterr3,filterang2,filterang3); 
figure; 
show(p1y,p2y); 
hold on; 
angy=[filterang1,filterang2,filterang3,filterang4]; 
ry=[filterr1,filterr2,filterr3,filterr4]; 
polar(angy,ry,'x'); 
hold off;   
% Compute 
     mass=358.9; 
     coill=0.1564; 
     global p11; 
     global p22; 
     global pixvalue; 
     p11=p1y; 
     p22=p2y; 
     pixvalue=coill/coil; 
     y11=quadl('volume1',0,pi); 
     v1=2*y11*pi/3; 
     densityrr=mass/v1*0.001;%%% Right+Right 
     y22=quadl('volume2',0,pi); 
     v2=2*y22*pi/3; 
     densityll=mass/v2*0.001;%%% Left+Left 
     densityrl=2*mass/(v1+v2)*0.001;%%% Right+Left 
     value1=[densityrr,densityll,densityrl,intensity]; 
67 
APPENDIX B 
THE SOURCE CODE FOR NUMERICAL MODEL OF MODULATED 
 POWER METHOD  
 
68 
% Modulated heating method simulation 
clc; 
 clear; 
 load powerdata; 
%-------Material Property ---------------------------------- 
  S_b=5.67*10^-8; 
  K=76;   
  Cp=620;  
  Den=7905; 
  Emis=0.22; 
  ti=300;  % enviroment temp 
  R_sample=2; % unit: mm 
  ele_cond=5e6; 
  permea_vacuum=4*pi*10^-7; 
  skin=sqrt(2/(200e3*ele_cond*permea_vacuum); 
  h=R_sample*sin(10*pi/180); 
  h1=(R_sample-skin)*sin(10*pi/180); 
  V_heat=2*pi*h^2*(R_sample-h/3)- 2*pi*h1^2*(R_sample-skin-h1/3); 
  volume=4*pi*R_sample^3/3; 
  mass=Den*volume*(1e-9); 
  A=4*pi*R_sample^2*(1e-6); 
% Discritize time and space domain 
  dt=0.05; 
  tfinal=2; 
  Nr=20; 
  Ntheta=30; 
  dr=R_sample/(Nr); 
  dtheta=pi/2/(Ntheta-1); 
   Tmatrix=ones(Ntheta*Nr+1,1)*1915.1;   
    Eqr=zeros(Nr*Ntheta+1,1); 
    Eql=zeros(Nr*Ntheta+1,Nr*Ntheta+1); 
    hp=0; % Power indicator 
    for tt=dt:dt:tfinal, 
        tt 
        hp=hp+1; 
      % Inside node 
        for row=2:Ntheta-1, 
            for colum=1:Nr-1, 
               node=colum+(row-1)*Nr; 
               theta=pi/2-dtheta*(row-1); 
               r=colum*dr ; 
               Ar=(r)^2*sin(theta)*dtheta*pi/2*(1e-6); 
               Ar1=(r-0.5*dr)^2*sin(theta)*dtheta*pi/2*(1e-6); %Inside(m^2)                
Ar2=(r+0.5*dr)^2*sin(theta)*dtheta*pi/2*(1e-6); % Outside 
Aa1=r*sin(theta-0.5*dtheta)*pi*dr/2*(1e-6);     % Up 
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Aa2=r*sin(theta+0.5*dtheta)*pi*dr/2*(1e-6);     % Down 
               c1=Den*Cp*Ar*dr*(1e-3)/dt; 
               c2=-K*Aa1/(r*(1e-3)*dtheta); 
               c3=K*Aa2/(r*(1e-3)*dtheta); 
               c4=-K*Ar1/(dr*(1e-3)); 
               c5=K*Ar2/(dr*(1e-3)); 
                    Eql(node,node-Nr)=-c3; 
                    Eql(node,node)=c1-c2+c3-c4+c5; 
                    Eql(node,node+1)=-c5; 
                    Eql(node,node+Nr)=c2; 
                  if (theta>80*pi/180) & (r>R_sample-skin), 
                     Q=power(hp)/V_heat; 
                  else 
                     Q=0; 
                  end; 
                    Eqr(node)=c1*Tmatrix(node)+Q*Ar*dr; 
    if row==2   % Because the center located node-1 position 
                    Eql(node,end)=c4; 
                  else 
                    Eql(node,node-1)=c4; 
                  end 
           end 
         end 
    % Surface node 
       colum=Nr; 
        for row=2:Ntheta-1, 
            node=colum+(row-1)*Nr; 
            theta=pi/2-dtheta*(row-1); 
            r=colum*dr; 
            Aa1=r*sin(theta-0.5*dtheta)*pi*dr/2*(1e-6)/2;      
            Aa2=r*sin(theta+0.5*dtheta)*pi*dr/2*(1e-6)/2;      
            Ar=(r)^2*sin(theta)*dtheta*pi/2*(1e-6); 
            Ar1=(r-0.5*dr)^2*sin(theta)*dtheta*pi/2*(1e-6);          
ha=S_b*Emis*(Tmatrix(node)^2+ti^2)*(Tmatrix(node)+ti);                              
c1=Den*Cp*Ar*dr*(1e-3)/dt; 
               c2=-K*Aa1/(r*(1e-3)*dtheta); 
               c3=K*Aa2/(r*(1e-3)*dtheta); 
               c4=-K*Ar1/(dr*(1e-3)); 
               c5=-ha*Ar; 
                    Eql(node,node-Nr)=-c3; 
                    Eql(node,node)=c1-c2+c3-c4-c5; 
                    Eql(node,node+Nr)=c2; 
                    Eql(node,node-1)=c4; 
                 if (theta>80*pi/180) , 
                     Q=power(hp)/V_heat; 
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                  else 
                     Q=0; 
                  end; 
                    Eqr(node)=c1*Tmatrix(node)-c5*ti+Q*Ar*dr;  
end 
     % Boundary nodes 
       row=1; 
           for colum=1:Nr, 
             node=colum+(row-1)*Nr; 
             Eql(node,node)=1; 
             Eql(node,node+Nr)=-1; 
           end 
           row=Ntheta; 
           for colum=1:Nr, 
             node=colum+(row-1)*Nr; 
             Eql(node,node)=1; 
             Eql(node,node-Nr)=-1; 
           end 
       % Center 
          Eql(Nr*Ntheta+1,Nr*Ntheta+1)=1; 
          Eql(Nr*Ntheta+1,1)=-1; 
        % Solve 
        Tmatrix=inv(Eql)*Eqr; 
        center(hp)=Tmatrix(1); 
        side(hp)=Tmatrix(Nr); 
end 
%function power=getpower(pos,I) 
  f= 200000; 
  sample_ID=1;  
[permea_vacuum,s_b,permea,permea_,conduc,emmis,R_sample,Density]=getsample(sa
mple_ID); 
    coil_ID=1;  
    [n_upper,n_lower,span,d,R_coil]=getcoil(coil_ID);    
    coillength=(n_lower-1)*d+(n_upper-1)*d+span; 
    % Calculate the levitation force 
          % Calculate Gx,Fx 
            x=R_sample*sqrt(pi*conduc*f*permea_); 
            xx=2*x; 
            if (x>1000), 
                Gx=1.0; 
                Fx=x; 
            else 
             temp1=3*(sinh(xx)-sin(xx)); 
             temp2=4*x*(sinh(x)^2+sin(x)^2); 
             Gx= 1-temp1/temp2;           
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Fx=(x*sinh(xx)+x*sin(xx)-cosh(xx)+cos(xx))/(cosh(xx)-cos(xx)); 
end; 
  % Calculate the item1 ~ item4 
          z=pos/1000; 
       for n=1:1:n_lower, 
              z_l(n)=-(span/2+(n-1)*d); 
          end; 
          z_l=rot90(z_l,2); 
          for n=1:n_upper, 
              z_u(n)=span/2+(n-1)*d; 
          end; 
          item1=0; 
          item2=0; 
          item3=0; 
          item4=0; 
          for n=1:n_lower, 
              M=R_coil^2; 
              temp1=(z-z_l(n))^2; 
              temp2=(z-z_l(n)); 
              temp3(n)=M/((M+temp1)^1.5); 
              temp4(n)=(M*temp2)/((M+temp1)^2.5);      
          end; 
          item3=sum(temp3); 
          item4=sum(temp4); 
          for n=1:n_upper, 
              M=R_coil^2; 
              temp1=(z-z_u(n))^2; 
              temp2=(z-z_u(n)); 
              temp5(n)=M/((M+temp1)^1.5); 
              temp6(n)=(M*temp2)/((M+temp1)^2.5);        
          end; 
          item1=sum(temp5); 
          item2=sum(temp6);       
         F_upper=9/8*permea_*I*I*Gx*item1*item2;  
           F_upper=F_upper/9.8/1000;             % (g/cm^3) 
         p_upper=3*pi*R_sample*(0.5*I*item1)^2*Fx/conduc; 
         F_lower=9/8*permea_*I*I*Gx*item3*item4;  
           F_lower=F_lower/9.8/1000;             % (g/cm^3) 
         p_lower=3*pi*R_sample*(0.5*I*item3)^2*Fx/conduc; 
         F=F_lower+F_upper; 
         p=p_upper+p_lower; 
      power=p; 
function force=getforce(pos,I2) 
  I=I2; 
   f= 200000; 
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   % Analytical Model of EML  
   % Load the thermophysical properties of the sample and sample size 
    sample_ID=1; 
[permea_vacuum,s_b,permea,permea_,conduc,emmis,R_sample,Density]=getsample(sa
mple_ID); 
    coil_ID=1;  
    [n_upper,n_lower,span,d,R_coil]=getcoil(coil_ID);    
    coillength=(n_lower-1)*d+(n_upper-1)*d+span; 
           x=R_sample*sqrt(pi*conduc*f*permea_); 
            xx=2*x; 
            if (x>1000), 
                Gx=1.0; 
                Fx=x; 
            else 
             temp1=3*(sinh(xx)-sin(xx)); 
             temp2=4*x*(sinh(x)^2+sin(x)^2); 
             Gx= 1-temp1/temp2;    
Fx=(x*sinh(xx)+x*sin(xx)-cosh(xx)+cos(xx))/(cosh(xx)-cos(xx)); 
           end; 
           z=pos/1000; 
       for n=1:1:n_lower, 
              z_l(n)=-(span/2+(n-1)*d); 
          end; 
          z_l=rot90(z_l,2); 
          for n=1:n_upper, 
              z_u(n)=span/2+(n-1)*d; 
          end;  
          item1=0; 
          item2=0; 
          item3=0; 
          item4=0; 
          for n=1:n_lower, 
              M=R_coil^2; 
              temp1=(z-z_l(n))^2; 
              temp2=(z-z_l(n)); 
              temp3(n)=M/((M+temp1)^1.5); 
              temp4(n)=(M*temp2)/((M+temp1)^2.5); 
              
          end; 
          item3=sum(temp3); 
          item4=sum(temp4); 
          for n=1:n_upper, 
              M=R_coil^2; 
              temp1=(z-z_u(n))^2; 
              temp2=(z-z_u(n)); 
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              temp5(n)=M/((M+temp1)^1.5); 
              temp6(n)=(M*temp2)/((M+temp1)^2.5); 
               
          end; 
          item1=sum(temp5); 
          item2=sum(temp6);   
         F_upper=9/8*permea_*I*I*Gx*item1*item2;  
         F_upper=F_upper/9.8/1000;             % (g/cm^3) 
         p_upper=3*pi*R_sample*(0.5*I*item1)^2*Fx/conduc; 
         F_lower=9/8*permea_*I*I*Gx*item3*item4;  
           F_lower=F_lower/9.8/1000;             % (g/cm^3) 
         p_lower=3*pi*R_sample*(0.5*I*item3)^2*Fx/conduc; 
         F=F_lower+F_upper; 
         p=p_upper+p_lower; 
      force=F; 
function  
[permea_vacuum,s_b,permea,permea_,conduc,emmis,R_sample,Density]=getsample(sa
mple_ID) 
          permea_vacuum=4*pi*10^-7; 
          s_b=5.67*10^-8; 
          permea=1;  
          permea_=permea*permea_vacuum; 
          conduc=3.85e6;   % Iron Sample 
          emmis=0.34; 
          R_sample=0.002;   
          Density=7.8;   %% g/cm^3 
function k=stiffspring(I,bottom,top) 
 f= 200000; 
   % Analytical Model of EML  
   % Load the thermophysical properties of the sample and sample size 
    sample_ID=1; 
[permea_vacuum,s_b,permea,permea_,conduc,emmis,R_sample,Density]=getsample(sa
mple_ID); 
    coil_ID=1;  
    [n_upper,n_lower,span,d,R_coil]=getcoil(coil_ID);    
    coillength=(n_lower-1)*d+(n_upper-1)*d+span; 
    % Calculate the levitation force 
          % Calculate Gx,Fx 
            x=R_sample*sqrt(pi*conduc*f*permea_); 
            xx=2*x; 
            if (x>1000), 
                Gx=1.0; 
                Fx=x; 
            else 
             temp1=3*(sinh(xx)-sin(xx)); 
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             temp2=4*x*(sinh(x)^2+sin(x)^2); 
             Gx= 1-temp1/temp2;        
Fx=(x*sinh(xx)+x*sin(xx)-cosh(xx)+cos(xx))/(cosh(xx)-cos(xx)); 
           end; 
        h=1; 
        low_limit=bottom;  
         up_limit=top; 
      dz=coillength/500; 
         for z=low_limit:dz:up_limit, %  Sample's position  
             
          for n=1:1:n_lower, 
              z_l(n)=-(span/2+(n-1)*d); 
          end; 
          z_l=rot90(z_l,2); 
          for n=1:n_upper, 
              z_u(n)=span/2+(n-1)*d; 
          end;             
          item1=0; 
          item2=0; 
          item3=0; 
          item4=0; 
          for n=1:n_lower, 
              M=R_coil^2; 
              temp1=(z-z_l(n))^2; 
              temp2=(z-z_l(n)); 
              temp3(n)=M/((M+temp1)^1.5); 
              temp4(n)=(M*temp2)/((M+temp1)^2.5);     
          end; 
          item3=sum(temp3); 
          item4=sum(temp4); 
          for n=1:n_upper, 
              M=R_coil^2; 
              temp1=(z-z_u(n))^2; 
              temp2=(z-z_u(n)); 
              temp5(n)=M/((M+temp1)^1.5); 
              temp6(n)=(M*temp2)/((M+temp1)^2.5);  
          end; 
          item1=sum(temp5); 
          item2=sum(temp6); 
            F_upper(h)=9/8*permea_*I*I*Gx*item1*item2;  
           F_upper(h)=F_upper(h)/9.8/1000;             
p_upper(h)=3*pi*R_sample*(0.5*I*item1)^2*Fx/conduc; 
         F_lower(h)=9/8*permea_*I*I*Gx*item3*item4;  
           F_lower(h)=F_lower(h)/9.8/1000;             
p_lower(h)=3*pi*R_sample*(0.5*I*item3)^2*Fx/conduc; 
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         F(h)=F_lower(h)+F_upper(h); 
         p(h)=p_upper(h)+p_lower(h); 
         h=h+1; 
     end; 
       Site=low_limit:dz:up_limit; 
       volum=4*pi*(100*R_sample)^3/3; 
       F=F*volum*9.8/1000; 
       [k,s]=polyfit(Site,F,1); 
       Ffit=polyval(k,Site); 
       figure; 
       plot(Site,F,Site,Ffit); 
       legend('Levitation','Fitted'); 
       ylabel('Levitation force N'); 
       xlabel('Position (m)'); 
76 
APPENDIX C 
THE SOURCE CODE FOR ANALYSIS OF TEMPERATURE DATA 
77 
clc; 
clear; 
sample=50;%% Sampling rate 
dataset=xlsread('Da'); 
dataset=dataset(1:2:end,:); 
time=dataset(:,1); 
temp=dataset(:,2); 
power=dataset(:,3); 
M=[time,temp,power]; 
%Fit time range 
 time1=300; 
 time2=410; 
 data=cutime(time1,time2,time,temp,power); 
 % Display 
 time=data(:,1); 
 temp=data(:,2); 
 power=data(:,3); 
 % 
 subplot(2,1,1), 
 plot(time,temp); 
 xlabel('time(s)'); 
 ylabel('Temperature (C)'); 
 subplot(2,1,2); 
 plot(time,power); 
 ylabel('Control voltage(V)'); 
  % 
 shift=mean(temp); 
 temp1=temp-shift; 
 [b,a] = butter(5,1/(sample/2)); 
 Hd = dfilt.df2t(b,a);         
 fitemp= filter(Hd,temp1)+shift; 
 % 
 point=100; 
 fitemp=fitemp(point:end); 
 power=power(point:end); 
 time=time(point:end)-time(point); 
  figure; 
 subplot(2,1,1); 
 plot(time,fitemp); 
 xlabel('time(s)'); 
 ylabel('Temperature (C)'); 
 subplot(2,1,2); 
 plot(time,power); 
 ylabel('Control voltage(V)'); 
 %xlswrite('Noisefree',[time,fitemp,power]); 
78 
% Data Analysis 
 clc; 
 clear; 
 numerical= xlsread('Noisefree'); 
 time=numerical(:,1); 
 temp=numerical(:,2); 
 power=numerical(:,3); 
 %% Display 
 subplot(2,1,1); 
 plot(time,temp); 
 ylabel('Temperatur (C)'); 
 subplot(2,1,2); 
 plot(time,power); 
 xlabel('time(s)'); 
 ylabel('Control voltage(V)'); 
 % curve fitting  
   % DC 
   time1=160; 
   time2=180; 
   temp_dc=findmean(time,temp,time1,time2); 
   %% Amplitude 
   %% Phase angle 
   time1=160; 
   time2=180; 
   f=0.15; 
   [ampli,dc,phase]=sinfit(time,temp,time1,time2,f); 
function [ampli,dc,phase]=sinfit(time,temp,time1,time2,f) 
 % Trim data 
  index1=find(time>=time1); 
  index1=index1(1); 
  index2=find(time>=time2); 
  index2=index2(1); 
  temp=temp(index1:index2); 
  time=time(index1:index2); 
   dcini=(max(temp)+min(temp))*0.5; 
   acini=(max(temp)-min(temp))*0.5; 
   % Phase 
   w=2*pi*f; 
   h=0; 
    for theta=0:pi/180:2*pi, 
        fitemp=dcini+acini*sin(w*time+theta); 
        h=h+1; 
        subtr=abs(fitemp-temp); 
        residual(h)=sum(subtr); 
        thetar(h)=theta; 
79 
    end 
    [record,index]=min(residual); 
    phase=thetar(index); 
      for dc=dcini-2:0.1:dcini+2, 
         for ampli=acini-1:0.1:acini+1, 
         fitemp=dc+ampli*sin(w*time+phase);      
          h=h+1; 
         subtr=abs(fitemp-temp); 
         residual(h)=sum(subtr); 
         dcr(h)=dc; 
         amplir(h)=ampli; 
         end 
     end 
      [record,index]=min(residual); 
      dc=dcr(index); 
      ampli=amplir(index); 
   % Display 
    fitemp=dc+ampli*sin(w*time+phase); 
    plot(time,temp,time,fitemp,'--'); 
    xlabel('time(s)'); 
    ylabel('Temperatur (C)'); 
    legend('Original data','fitted data');