Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013 (50–55)

Theorization and Modeling Ⅲ(Review)

Electromagnetic Levitation – White-noise Measurement Protocol for
Modulated Calorimetry
Jacqueline ETAY
Abstract
Electromagnetic levitation of electrically conductive droplets by alternating magnetic fields is a technique used to measure the physicalproperties of liquid metallic alloys as heat capacity and thermal diffusivity, among others. In order to reduce electromagnetic stirring andshaping of the molten sample, experiments are conducted in microgravity 1). Measurement is performed by using the modulation calorimetry
technique [2][3]. Here we test both numerically and experimentally a new measurement protocol, which aim is to eliminate calibration. We use
this procedure to demonstrate that the use of a levitator in a DC magnetic field overcomes the negative effects of the electromagnetic stirringinside the sample on the accuracy of the modulated calorimetry technique.
Keyword(s): Electromagnetic levitation, modulated calorimetry, identification, white-noise, magnetic fields superposition

1. Introduction
Numerical models of casting and solidification are
increasingly used to optimize industrial processes for the
production of metal alloys. All such models require, as input
parameters, reliable thermophysical property values such as heat
capacity and thermal conductivity. However, most standard
measurement techniques, such as differential scanningcalorimetry, require a molten metallic sample to be in contact
with a crucible, leading to unavoidable contamination of thissample. Based on the early work of Okress 4) on electromagnetic(e.m.) levitation, Egry, see for example 5), developed anelectromagnetic levitator (EML), a non-contact technique,which when used in microgravity conditions, reduces flows
inside the molten sample. This instrument is used in theframework of the Thermolab ESA-MAP project (Fecht 6)).
In this article, we first present how the experimental device
works. Then we briefly sum up the modulation calorimetry
technique where thermal behavior of levitated sample isdeduced from unsteady thermal simulation.
In the present work, signals are processed by a signal analysis
technique coming from Process Engineering (Schetelat 7) ). It
couples both a white-noised modulated input power and an
identification procedure. This procedure allows to derivate
relevant transfer functions of the considered system. The
eigenfrequencies of those functions are relevant from external
and internal heat exchanges of the sample.
First, calculated time-dependant temperatures are used as
experimental recorded signals. This sequence is performed
numerically both on solid and liquid, both varying viscosity andintensity of a superposed DC transverse magnetic field.
Electromagnetic stirring effects on heat transfer are deduced
SIMAP/EPM Phelma Campus BP75 38402 St Martin d’Heres Cedex(E-mail : jacqueline.etay@grenoble.cnrs.fr)

from the comparison between solid and liquid results. Analysisof the obtained results gives clear insights on how to manage
modulated calorimetry to find reliable value of both heat
capacity and thermal conductivity.
Second, we present experimental tests performed on a solid
niobium bar. Experimental and literature results are compared.

2. Measuring principle
The principle of the measure is given by Fecht 2) and byWunderlich 3). It is sketched on Fig.1.
A spherical sample of radius R and electrical conductivity  islocated in an inductor powered by a high frequency alternating
current: 𝐼𝑜 𝑐𝑜𝑠𝜔1 𝑡, 𝐼𝑜 and  being the peak current and theangular frequency respectively. This inductor generates amagnetic field which induces, inside the sample, electricalinside the inductor and system representation of thearrangement (inductor+sample) currents localized in asurface layer called the electromagnetic skin depth. The effects

Fig. 1 Schematic representation of the sample.
FRANCE− 50 −

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Jacqueline ETAY

of these currents are twofold. First, they generate a Joule heating
power P inside the sample. Second, they compose with themagnetic field to create a body force field able to center (orlevitate) the sample.
When the sample is liquid, those body forces generate
electromagnetic stirring. In the molten sample, the resulting
velocity field evolves as the Alfven velocity - given by thebalance between inertia and electromagnetic forces term in the
momentum equation - 𝑈𝐴 ∝ (𝐼𝑜 ⁄𝑅)√𝜇⁄𝜌, where 𝜌 and 𝜇 are
the density of the sample and the permittivity of the vacuum
respectively. A Reynolds number, defined as, 𝑅𝐴 = 𝑈𝐴 𝑅⁄𝜈
where 𝜈 is the kinematic viscosity, will be used to characterize
the electromagnetic stirring.
Moreover, as proposed by Bojarevics 8) to damp
electromagnetic stirring motion, a transverse uniform
continuous magnetic field 𝐵𝐷𝐶 can be applied. The nondimensional number characterizing the effect of this DC field isthe Hartman number 𝐻𝑎 = 𝐵𝐷𝐶 𝑅√𝜎⁄𝜌𝜈. Note that, such a tool
is not yet set up on a real levitation device.
Modulated calorimetry consists in perturbing the thermal
equilibrium of the sample by modulating the total Joule power P
around its mean value 𝑃̅ (𝑃 = 𝑃̅ + 𝑃̃ ) and recording therelated time-dependant behavior of surface temperatures, i.e.
̃𝑝 (𝑡) and 𝑇𝑒 (𝑡) = 𝑇̅𝑒 + 𝑇̃𝑒 (𝑡) the polar and the𝑇𝑝 (𝑡) = ̅̅̅
𝑇𝑝 + 𝑇equatorial temperatures. Currently, an analytical model 2) is used
and writes as :
𝐶𝑝 𝑔𝑒

̃𝑒
𝑑𝑇
𝑑𝑡

̃𝑝 − (1 + 𝑠𝑒 𝐵𝑖)𝑇̃𝑒 ] + 𝑃̃
= ℎ𝑖𝑛𝑡 [𝑇𝐶𝑝 (1 − 𝑔𝑒 )

̃
𝑑𝑇
𝑝
𝑑𝑡

̃𝑝 ]
= ℎ𝑖𝑛𝑡 [𝑇̃𝑒 − (1 + (1 − 𝑠𝑒 )𝐵𝑖)𝑇

(1)
(2)

where se, ge, hint, and Bi are respectively the surface area and thevolume ratios of the equatorial zone, the internal global heattransfer coefficient and the Biot number.
It is established that for a well-tuned 𝜔2 modulation frequencyof the Joule input power, 𝑃̃ = 𝑃𝑜 𝑐𝑜𝑠(𝜔2 𝑡) , 𝐶𝑝 and𝜅𝑡ℎ respectively the heat capacity and the thermal conductivity
can be calculated by using the following equations
𝐶𝑝 =

𝑃𝑜

1

̃
𝜔2 𝑇
𝑝 [1+(𝑔𝑒 (1−𝑠𝑒 )+𝑠𝑒 (1−𝑔𝑒 ))𝐵𝑖]

𝜅𝑡ℎ =𝐶𝑝 𝑔𝑒 (1−𝑔𝑒 )𝜔22
4𝜋(𝑅−𝛾𝛿)𝜔𝑒𝑥𝑡

(3)
(4)

where 𝛾and 𝛿 are respectively a geometrical factor and theelectromagnetic skin depth.
To use this model, calibrations are needed to establish thevalues of se and ge the geometrical factors and 𝜔𝑒𝑥𝑡 thecharacteristic angular frequency of the heat transfer from thesample to its environment.
The measuring system is sketched on Fig. 1-top when on thebottom a system representation of the responses to themodulated power is presented. The functions 𝐻𝑝 and 𝐻𝑒 are
the transfer functions relative to the pole and equatorial

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Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

Fig. 2quasistationnary

capacitive

ext

int

low
frequency

intermediate
frequency

conductivehighfrequency

Characteristic thermal behavior in a solid
sample versus the modulation frequency
value.

temperature behaviors respectively.
To illuminate how the sample should reacts to different
modulation frequencies 𝜔2 , we draw, on Fig. 2, a schematic
thermal behavior of the sample versus 𝜔2 . It is defined as theratio of the amplitude of the temperature fluctuation 𝑇̃𝑝 on theamplitude of the power fluctuation 𝑃̃. For low frequency values𝜔2 , the heat exchange between the sample and the exterior isdominating, leading to an almost stationary regime. Therefore
the time variation of the sample temperature follows the time
variation of the dissipated Joule power leading to a flat spectrum.
For very high frequencies, that is to say greater than a criticalvalue 𝜔𝑖𝑛𝑡 , temporal variations of the temperature variations do
not follow power variations anymore, the slope of the spectrum
changes. When the sample is solid, it is a conductive regime andthe slope is found to be equal to 𝜔𝑖𝑛𝑡 ⁄𝐶𝑝 𝜔22 . For intermediate
values, both internal and external heat exchanges control thetemperature field. This is the range to choose the modulation
frequency of the input power, when it is harmonic, i.e.
depending on one frequency only 3). As proposed by Schetelat 7),
using a white-noised input power, allows the experimenter toinvestigate the complete thermal behavior of the sample, giving
him dispense to the knowledge of the geometry of this sample
and to the choice of a specific modulation angular frequency.
Such spectrum allows us to find 𝜔𝑒𝑥𝑡 and 𝜔𝑖𝑛𝑡 two natural,also called eigen-frequencies calculated as the frequencies atwhich the slope of the spectrum changes. 𝜔𝑒𝑥𝑡 is inverse of thecharacteristic time of heat transfer between the sample and thesurrounding atmosphere, when 𝜔𝑖𝑛𝑡 is related to the time oftransfer inside the sample. Therefore, the 𝜔𝑖𝑛𝑡 value will
possibly be affected by the flow. We will show that this value
can be use to find 𝐶𝑝 and 𝜅𝑡ℎ attached to a solid sample or toa liquid sample when the EML is located in a DC magnetic field.

3. Simulation – Geometry and Software
To test the procedure outlined in the introduction section, wechoose the geometry of Fig. 3 and the simulation parameters
given in Table 1. These geometry and parameters are chosen− 51 −Electromagnetic Levitation – White-noise Measurement Protocol for Modulated Calorimetry

Table 1

Simulations parameterssample radius 𝑅, mm
density 𝜌 kg.m-3
heat capacity 𝐶𝑝 , J.K-1.kg-1
thermal conductivity 𝜅𝑡ℎ  W.m-1.K-1
electrical conductivity 𝜎𝑒𝑙 , -1.m-1
total hemispherical emissivity 𝜀inductor current 𝐼𝑜 , Apeak
currents angular frequency 𝜔1 , rad.s-1

4
3860
663
23.1
5.26*1050.4
652*350*103

Fig. 3 Sample and inductor geometries

because they are representative of the on-board EML inparabolic flights, sounding rockets and the International Space
Station. All physical properties are kept constant, i.e. notdepending on temperature.
To emulate the temperature signals, we use the Fluent
software in which Delannoy 9) added a modulus that describes
induction.
For those calculations, the Alfven velocity is fixed to
𝑈𝐴 = 0.3733 𝑚. 𝑠 −1 . We choose conditions such that the flows
remind laminar. Indeed, we know that the presence of
turbulence would prevent measurements. We will go back to thisassumption in the concluding section. The upper DC magnetic
field intensity is limited to 1.2 Teslas by the used mesh.
Surface velocities, normalized by the Alfven velocity andcalculated under various viscosity conditions are plotted on Fig.
4. On the top, the Reynolds number 𝑅𝐴 varies, when, on thebottom, the intensity of the DC applied magnetic field varies,
the liquid viscosity being fixed at 𝜈𝑜 = 10−6 𝑚2 . 𝑠 −1 .
As expected, calculated flows are symmetrical about theequator ( = 90°). Those shapes are corresponding to two
vortices in the bulk flow. The turnover angular frequency of theFig. 4

Fig. 5

Calculated velocity on the surface of aspherical sample varying ReA (top) and Ha(bottom).

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013− 52 −

Polar transfer function calculated byprocedure proposed in 8) and corresponding
to motions presented on Fig.4- top : varying
viscosity – bottom : varying intensity of theDC magnetic field applied transversally.

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Jacqueline ETAY

bulk flow is calculated as 𝜔𝑣𝑜𝑟𝑡𝑒𝑥 = 〈∇ × 𝑈〉⁄2 . Comparing
both the left and right graphs, we see they are quite similar.
Nevertheless, the point where the surface velocity is maximum

Table 2 Angular eigen-frequencies of the polartransfer function plotted on Fig. 5 comparedwith the turnover frequencies of the e.m.
stirring vortex.

is closer to the equator ( = 90°) in the presence of an applied
DC magnetic field than in its absence. In such a case, this point orange is  = 45° and 135° (for 𝑅𝑒𝐴 = 16) and  = 50° and
130° (for 𝑅𝑒𝐴 = 80).
How the flow changes the temperature field has been reported
by Schetelat 10). His conclusions were that, on the one hand,convection homogenizes the temperature field in the sample
bulk and on the other hand, when active enough, it causes atemperature inversion near the equator. This generates aparadoxical phenomenon: in the equatorial zone where the input
Joule power density is maximal and the temperature the lowest.
Once we have emulated the power and the corresponding
velocity field, we use a pseudo white noise signal to modulate
the power. It is generated by a Linear Feed Back Shift Register,
commonly used in electronics and communication engineering.
Then recorded temperatures are analyzed by an identification
procedure. It allows the determination of 𝐻𝑝 and 𝐻𝑒 theInt. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

ext

int

68.13

0.2112

65.80

40

39.3

35.70

0.2107

38.05

75

21.0

19.24

0.2101

22.87

100

15.7

14.72

0.2099

18.71

solid

0

0

0.2076

6.54

BDC

T 

Ha

vortex

0.5

24

0.6

rd.s 
1

Hp
ext

int

164.050.2113162.9028.7

124.050.2113123.33

0.8

38.3

81.16

0.2113

79.53

1

47.9

57.58

0.2112

59.34

1.2

57.9

41.36

0.2110

48.34

solid

-

0

0.2076

6.54

0.6
U U A max

0.8

4. Results

53

Hp

78.6

and 𝐻𝑝 are to equal 𝜔𝑒𝑥𝑡 and 𝜔𝑖𝑛𝑡 .

inttransfer function. The black curve is attached to the solid andacts as the reference. From those curves, the eigen-frequenciesof transfer function are calculated and reported in Table 2. Inorder to compare with a value attached to the flow itself, we
introduce 𝜔𝑣𝑜𝑟𝑡𝑒𝑥 the turnover angular frequency of thevortices as well.
We see that the external angular frequency 𝜔𝑒𝑥𝑡 variation isless than 2%. Therefore, we are allow to affirm that theexistence of the flow does not significantly modify the heatexchanges between the sample and the surrounding atmosphere(in the present case radiation only). 𝜔𝑖𝑛𝑡 strongly evolves withthe flow in a very similar way than 𝜔𝑣𝑜𝑟𝑡𝑒𝑥 . Comparing thosevalues leads us to affirm that higher 𝑅𝑒𝐴 is, more similar 𝜔𝑖𝑛𝑡and 𝜔𝑣𝑜𝑟𝑡𝑒𝑥 are. The difference between those values is thehighest for solid (i.e. vanishing velocity). Therefore, we may
affirm that the heat transfer inside a molten sample is driven, forlow velocity by both conduction and convection, and for highvelocity by convection only. This result could appear to bestraightforward, but it is not when considering the systemapproach chosen to analyze recorded temperature signals only.
Now, on Fig. 6, we plot the 𝜔𝑖𝑛𝑡 values reported in Table 2
versus the |𝑈|𝑚𝑎𝑥 ⁄𝑈𝐴 values deriving from the Fig. 4 graphs.
These points are in line. The regression line crosses the y-axis inrd.s 
1

20transfer functions of the sample temperature variation due toheating power variation. The eigen-frequencies values of 𝐻𝑒

Results corresponding to the flow reported on Fig. 4 areplotted on Fig. 5. They are the spectra of 𝐻𝑝 the polar heatvortex

Re A

0.0

0.2

0.4

1.0

1.2

Internal angular eigen-frequency versus
maximum velocity attached to Fig. 4
tiles : variation due to 𝑅𝑒𝐴 – squares : due to 𝐻𝑎.
Fig. 6the point corresponding to a vanishing velocity. Therefore, we
may propose a new strategy for measuring both heat capacity
and thermal conductivity of liquid metal alloy joining
electromagnetic levitation plus a DC magnetic field and the here
presented measurement protocol.
Measurements would be performed for two (or three) highenough magnetic field intensity values. The internal eigen-− 53 −Electromagnetic Levitation – White-noise Measurement Protocol for Modulated Calorimetry

frequencies would be calculated for each value and be plottedversus those values. Then the straight line connecting thesepoints would be drawn. The point where this line crosses the yaxis is the eigen-frequency attached to the solid sample (i.e.
thermal conductivity is controlling internal heat transfer) Then,on this eigen-frequency, simply apply a current calorimetry
method as reported in 3) or an inverse method to obtain both heat
capacity or thermal conductivity.

5. Experimental validation
The experimental facility is sketched on Fig. 7. It is astandard levitation facility described in 10).
For the first trials, we chose a geometry as simple as possible:
the sample is solid and as such that its thermophysical properties
are known in the investigated temperature range. It is placed ona thermal insulating support and the inductor is similar to the
inductor used in for space experiments and corresponds to Fig. 3.
Power is modulated by a suitable white noise. Two dual band
pyrometers are arranged on the axis and in the plane of theinductor. Signals collected by these pyrometers are treatedaccording to the procedure described in 7) implemented in a
Matlab environment. In the examples below, the protocol isapplied to a niobium bar (length : 100 mm and diameter : 10mm).
Figure 8 shows a result of the identification of the measured
temperatures for the following experimental conditions: average
inducting current: 211 A - amplitude of the modulation of theinducting current: 8% - base frequency of the inducting current:
94,195 Hz. The parameters for the LFSR are chosen as specified
in annex 1 of reference (7) : 𝑁 = 7, 𝑘 = 1, 𝑡𝑠 = 4𝑠 - total
duration of the experiment: 500 𝑠
On Fig. 8, we may see that the recorded signal are wellrecovered by the identification procedure. This procedure isapplied for several inducting current intensities.
Figure 9 is the representation of 𝜏𝑒𝑥𝑡 = 1⁄𝜔𝑒𝑥𝑡 the evolutionof the characteristic time of heat transfer between the sample
and its environment versus the average surface temperature.

We note that this time is intrinsic of the transfer functions
𝐻𝑝 and He. It decreases with temperature. This trend is expected
as the radiative transfers increase with the temperature of thesurface of the sample.
The lower transfer time is 𝜏𝑒𝑥𝑡 = 132 𝑠 . By using thedefinition of this transfer time given by Fetch [2] i.e.
3
𝐶𝑝 ⁄4𝐴𝜀𝜎𝐵 𝑇𝑚𝑜𝑦
(where A and 𝑇𝑚𝑜𝑦 are respectively the area
and the mean temperature at the surface of the bar, we fund anemissivity 𝜀 = 0.996. Taking into account that the used bar isoxidized, this value is a good order of magnitude of theemissivity.
Figure 10 is the representation of the evolution of the heat
capacity, calculated by Eq. (3) versus the mean temperature of
the surface of the sample for both literature [11] andexperimental values. There is a good agreement for the points
corresponding to the highest investigated temperatures. The
discrepancy attached to the lower temperature points could be
due to the pyrometer accuracy. Indeed, the used pyrometers are
not so precise in the range of temperature lying between 750
and 850 °C.

Fig. 8 Identification of temperature variations
generated by a white noise modulation of theJoule power (recorded blues curve-identified
red curves) for above values - The upper andlower curves represent respectively the timevariation of equatorial temperature Te andthe polar temperature Tp.

Fig. 9 Evolution of ext the time characteristic of theheat transfers between the sample and its
environment. This time is model free.

Fig. 7 Experimental arrangement

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013

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Jacqueline ETAY

calculate the values of heat capacity and thermal conductivity ofthe sample.
A practical implementation of the procedure has beenachieved on as simple as possible facility. Comparisonsperformed on measured and literature heat capacities are goodenough to be sure that the used protocol is experimentally
validated.

Acknowledgement
This work has been funded by the followings supports : ESA
under MAP-Thermolab, EC under PI-IMPRESS (Contract
Number: NMP3-CT-2004-500635), CNES under Material
Sciences Program. Author thanks Dr Mazen Al Amir
(GipsaLab-Grenoble) for his invaluable help.

Fig. 10 Evolution of the heat capacity versus themean temperature of the surface of thesample. There is a good agreement for the
points corresponding to the highest
investigated temperatures.

References

6. Conclusion
In this article we presented some numerical experiments
leading us to propose an original method of measurement
derived from the traditional modulated calorimetry. It differsfrom existing methods on the following 3 points:- use of electromagnetic levitation joined to a continuous
magnetic field which intensity is high enough, as done by
Kobatake 12) but with different geometry,- use of a temporal modulation of the Joule power
dissipated into the sample in a white noise form,- use of a signal analysis coming from process engineering
to determine the eigen-frequencies attached to internal andexternal heat transfers.
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enough in order the flow to be laminar. At two values of appliedmagnetic field are attached two values of eigen-frequency of theinternal heat transfer. By interpolation of these two eigenfrequencies, we may find the frequency corresponding to thevanishing velocity. Knowing this point and using or ananalytical method 2) and 3), or an inverse method, we may

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Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

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(Received 9 Oct. 2012; Accepted 25 Dec. 2012)