MEASUREMENT MAGNETIC FORCE THAT BE
EXERCISED ON THE TUBE
TUBE WITH FERROFLUID
Corneliu BUZDUGA
Ştefan cel Mare University of Suceava
Abdelkader BENABOU
Polytech’Lille
REZUMAT. Acest articol prezinta măsurarea forţei magnetice ce se exercită asupra unui
unui tub cu lichid magnetic atunci când este aplicat
un câmp magnetic, abordând două modele. Primul model experimental, preia un experiment descris de R.E. Rosensweig, măsurând
inducţia magnetică B, intensitatea câmpului magnetic H şi forţa magnetică Fm, apoi
apoi calculând magnetizaţia ferofluidului după metoda
Rosensweig. După obţinerea acestor valori ss-a realizat trasarea grafică a caracteristicilor B(H), M(H) şi Fm(H). Al doilea model este o
simulare utilizânt Finite Element Method Magnetics (FEMM), apoi utilizând
utilizând funcţia lui Langevin calculăm magnetizaţia M, apoi trasăm
grafic aceleaşi caracteristici ca şi la modelul experimental.
Cuvinte cheie:
cheie ferofluid, forţă magnetică, inducţie magnetică.
ABSTRACT. This paper presented measurement the magnetic force exerted
exerted on a tube with magnetic fluid when applied magnetic
field, approaching the two models. The first experimental model, takes an experiment described by R.E. Rosensweig, measuring
magnetic induction B, magnetic field H and magnetic force Fm, then calculating
calculating the ferrofluid magnetization M by the method
Rosensweig. After obtaining these values was done drawing graphic characteristics B (H), M (H) and FM (H). The second model is
is
using the Finite Element Method Magnetics (FEMM) and using Langevin function calculate
calculate the magnetization M, then plot
graphically the same characteristics as the experimental model.
Keywords:
Keywords ferrofluid, magnetic induction, magnetic force.
1. INTRODUCTION
Ferrofluids are special materials with magnetic
properties, keeping properties the basic of liquids. This
material has three components: the liquid carrier can be
any liquid, ferromagnetic particles (Fe2O3) and a
stabilizer to prevent sedimentation and agglomeration
of particles (oleic acid). Ferromagnetic particles are
obtained by special methods (e.g. mechanical
dispersion) with a size of about 10 nm [6, 7, 10]
Magnetic
fluids
have
different
technical
applications, is controllable in the presence of a
magnetic field. By applying a variable magnetic field,
they may be forced to flow or can be kept suspended in
a fixed position.[3, 4, 5]
2. MODEL ROSENSWEIG
The Bernoulli equation is well known in
hydrodynamics and electromechanics. Recently its
scope was also extended to ferrohydrodynamics (FHD),
which, according to Rosensweig, seals with mechanics
of fluid motion influenced by strong forces of magnetic
polarization. The FHD Bernoulli equation introduced
by Rosensweig accounts for changes in magnetic
energy of a magnetizable fluid that in a magnetic field.
The experiment illustrated in figure 1 provides a
means for gravimetric measurement of the fieldaveraged magnetization. A tube of weight wt having
cross-sectional area at and containing ferrofluid is
suspended vertically by a filament between two poles
furnishing a source of applied field Ha. The top surface
of the ferrofluid at plane 1 experiences neglijible field
intensity, while at plane 2 the field H2 within the fluid
its uniform at maximum value. The force F is given as
the sum of pressure forces and weight of containing
tube [1, 2]:
F = p 2* − p 0 a t + wt
(1)
The generalized Bernoulli equation is:
(
p* +
)
−
1 2
ρv + ρgh − µ 0 M H = const
2
(2)
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where ρ is fluid density, g is the acceleration due to
gravity, h is the height of the fluid, µ0 is the
permeability of free space, M is the magnetization, H is
−
the applied field, M is the averaged magnetization
−
For a circular cylinder having length that exceeds its
diameter, the demagnetization coefficient D=1/2.
3. MODEL EXPERIMENTAL
H
∫
( M = (1 / H ) MdH ), p* is the combination of the
0
thermodynamic pressure, p, the magnetostrictive
pressure ps and pm, p * = p + p s + p m .
where p=p(ρ,T),
H
 ∂M 
ps = µ0 ∫υ 
 dH ,
∂
υ

 H ,T
0
(3)
and the fluid-magnetic pressure pm:
H
−
p m = µ 0 ∫ MdH = µ 0 MH
(4)
0
From the generalized Bernoulli equation applied
between section 1 and 2:
−
p1* + ρgh1 = p 2* + ρgh2 − µ 0 M H 2
(5)
p1* + p n = p0, 2 + pc , with
From the equation
After the above model proposed by Rosensweig, I
realized experimentally in a laboratory plant for
measuring the magnetic force. For this we used a
magnetic field generating device such Electroaimant
EA80F with variable 0-30 mm air gap, supplied from a
power type Albs JFA Electronique 3010, with 30V/10A
scale, a force sensor with a AC / DC Clamp Meter with
scale 400V AC / DC, 30A AC / DC, a GaussTeslameter FH-55 for induction and magnetic field
measurements and a tube of ferrofluid placed between
electromagnets. The principle scheme experiment is
shown in fig. 2.
Generation of magnetic field created by
electromagnets 1 and 1', they are power supply. Tube
made of plastic material has dimensions of 80 mm
height and diameter of 18 mm, the ferrofluid APG E32
type 2, has height 50 mm and is sealed with a tube 5.
pn=0 and pc=0, the boundary condition at the free
surface is p1* = p 0 .
Result:
−
M =
(F − F0 ) / at
µ0 H 2
(6)
where F0 = wt + ρg (h2 − h1 )a t , represents the forces
when the field is absent. H2 is less than applied field Ha
owning to the influence of the sample shape.
Introducing the definition of demagnetization
coefficient D ≡ (H a − H ) / M , in the present problem
gives H 2 = H a − DM .
Fig. 2. Experiment setup. 1, 1’ – electromagnets,
2 – ferrofluid, 3 – sensor of the force, 4 – fire for suspend tube, 5 –
tube, 6 – support, 7 – table, 8 – sensor for measuring magnetic field.
The force sensor with a scale up to 5 N, is suspended
tube containing ferrofluid, with a wire between the two
electromagnets, so that strength to be the highest value.
The values force a read by a Clamp-Meter 400V/30A
AC / DC to 1V ≈ 1N.
Fig. 1. Model R.E. Rosensweig. Reproduced after [1]
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Table 1. Values measurement
U[V]
I[A]
B[T]
H2[MA/m]
F0[N]
F[N]
1
3
5
7
9
11
13
0,5
1,3
2
2,8
3,7
4,5
5,3
0,037
0,100
0,161
0,228
0,293
0,357
0,420
0,030
0,080
0,129
0,181
0,232
0,285
0,335
2,1
2,1
2,1
2,1
2,1
2,1
2,1
2,2
2,5
2,8
3,1
3,4
3,7
4
Fm=FF0[N]
0,1
0,4
0,7
1
1,3
1,6
1,9
Fig. 3. Overview of the experiment stand
( F − F0 ) / at
M =
µ0 H 2
(6’)
B f = µ0 (H + M )
(7)
B f = µ0 µr H 2 ⇒ µr =
Characteristic B-H
0,5
0,4
0,3
0,2
0,1
0
0
Bf
µ0 H 2
where: permeability absolute is
Fig. 4. Overview of the experiment stand. Detail
B[T]
To have enough points to draw graphs, we measured
for several values of magnetic field, varying power
supply. The gravitational force in the absence of
magnetic field F0 = 2,1N and the maximum total force F
= 4N. Magnetic force exerted on the tube of ferrofluid
is Fm = F-F0. The measured values are presented in
table 1.
After the measurements we obtained the B-H
characteristic, presented in figure 5.
Calculation using the relations (6’),(7),(8), obtaining
magnetization M, induction magnetic in ferrofluid Bf
and the relative permeability µr.
µ 0 = 4π ⋅ 10 −7
at= 0,000254469 m3 - cross-sectional area
Results are presented in table 2.
100000
300000
400000
Fig. 5. Characteristic B-H experimental
Table 2. Values calculation
(8)
and
200000
H2[A/m]
M[A/m]
10429,277
15643,915
16977,893
17286,094
17531,974
17565,098
17745,337
Bf[T]
0,051
0,120
0,183
0,249
0,313
0,380
0,443
µr
1,348
1,196
1,132
1,096
1,076
1,062
1,053
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245
Magnetization characteristic is in figure 6.
Characteristic M(H)
20000
M[A/m]
15000
10000
5000
0
0
100000
200000
300000
400000
H2[A/m]
Fig.6. Magnetization characteristic experimental
Evolution of the magnetic function in report to
magnetic induction is presented in figure 7.
Characteristic Fm(B)
2
Fm[N]
1,5
Are used and two programs dedicated to display the
correct results:
Triangle.exe. Triangle geometry model divides the
whole into a large number of triangles, a vital step finite
element method.
Femmplot.exe. This small program used to display
graphics can also save and view bidimensional.
For the model I declare boundary conditions and
properties materials. In FEMM the Boundary Property
dialog box is used to specify the properties of line
segments or arc segments that are to be boundaries of
the solution domain. To simulation model boundary
conditions are variables. In properties materials, relative
permeability µr is fix for “Air” and “Fer”, for
“Ferrofluid” is variable [6].
I am solving problems on two-dimensional planar,
magnetostatic problem. First step I realized geometry
model with mesh, see figure 8 a). After execution
FEMM, distribution induction magnetic and force
magnetic see in figure 8 b).
1
0,5
0
0
0,1
0,2
0,3
0,4
0,5
B[T]
Fig.7. Characteristic Fm(B) experimental
4. MODEL FEMM
Finite Element Method Magnetics (FEMM) FEMM
is a suite of programs that solve problems magneto
statics frequency low. Programs currently solve
problems in two-dimensional and flat areas
axisymmetric. FEMM is divided into three parts:
Preprocessor (femme.exe). This is a CAD program
to achieve geometry model to define material properties
and to define boundary conditions. AutoCAD files can
be imported and the extensions to .dxf facilitate the
analysis of existing geometries in this format.
The solver (fkern.exe). Solver reads a set of data
describing the problem and solves Maxwell's equations
to obtain values of magnetic field in describing the
chosen field.
Postprocessor (femmview.exe). This is a program
that displays the resulting calculation of the magnetic
field lines form field or form of magnetic flux density.
Also, the program allows the user to observe the values
of different sizes are arbitrarily chosen magnetic points
and to evaluate different designs of sizes and full of
interest on a predefined shape [8].
a)
b)
Fig. 8. Geometry with mesh and distribution induction magnetic for
model in FEMM
Using MATLAB with OPTIMTOOL option and
OctaveFEMM we can make a simulation in an
automatic loop. [9]
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 cosh(αH )
1 
M = M sat ⋅ 
−

 sinh(αH ) αH 
OctaveFEMM is a set of tools that allow Matlab
program with a set of functions to work with files Finite
Element Method Magnetics (FEMM).
(12)
Using the optimization function, you can turn
fminsearch the syntax: x = fminsearch (fun, x0, options)
minimizes taking into account the parameters specified
in the structure optimization options, which are
generally specified by the user through optimset
function, these options are: display (may be 'off ', when
nothing appears on the display, 'iter', you see each
iteration, 'end', when the last iteration, or standard
option 'notify', when they give information whether the
solution not converge), TolX (error accepted for the
solution), MaxFunEval (maximum number of
evaluation functions assigned) and MaxIter (maximum
number of iterations you want).
Table 6. Values after optimization
B[T]
0
0,04
0,1033
0,1632
0,23
0,2953
0,3574
0,4211
Fig. 9. Logical scheme
When it launched OctaveFEMM execution FEMM
user interface is displayed during operation. The user
can choose work assignments, modeling and analysis,
whether solely through functions provided by the
toolbox, or a combination of manual and programmatic
operations.
The logical scheme is shown in figure 9.
For MATLAB algorithm, using Langevin function,
with formula:

 kT  1 
µ m
 ⋅ 
M = M sat ⋅ coth  0  H − 
kT
µ
m


0

 H

H[A/m]
0
29890
80132
129103
182200
233165
284129
334522
M[A/m]
0
2,31E+04
2,59E+04
2,69E+04
2,73E+04
2,76E+04
2,78E+04
2,80E+04
F[N]
0
0,0856
0,3978
0,6925
1,0036
1,3032
1,5907
1,889
After optimization we obtained of the characteristics
B(H), M(H) and Fm(B) presented in figures 10, 11 and
12.
(10)
where: M – magnetization, Msat – saturation
magnetization, µ0 – permeability free air, m – magnetic
moment of the particle, k – Boltzman constant, T –
temperature absolute, H – intensity magnetic field.
If Langevin function argument is α =
µ0m
kT
and we
Fig. 10. Characteristics B(H)
obtain:
1 

M = M sat ⋅ coth (αH ) −
αH 

(11)
or
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The ferrofluid used in the experiment is type APG
E32 and has the following specifications presented in
Table 3 [11].
Table 3. Ferrofluid technical specifications
carrier liquid
hydrocarbon
Ms[Gauss]
330
η[cP]
1200
ρ[g/cm3]
1,15
You can use various types of ferrofluids, but when
using ferrofluids with a higher magnetization, it is
necessary to apply a high magnetic field, which by
default will need a much stronger force sensor.
Fig. 11. Characteristic M(H)
Fig. 12. Characteristics F(B)
5. CONCLUSIONS
The methods described in this article have been used
to study ferrofluids and applied magnetic field on them.
If the experiment we used a plastic tube as a cylinder, to
continue this study we used a tube of glass or other
transparent material of different shapes such as
parallelepiped.
REFERENCES
R.E.
Ferrohydrodynamics,
Cambrige:
[1]. Rosensweig
Cambridge University Press, 1985.
[2]. Zimmels Y. The Bernoulli Equation for Fluids in
Electromagnetic and Interfacial Systems, Journal of Colloid
and lnterfaee Science, Vol. 125, No. 2, October 1988.
[3]. Odenbach
S.
Ferrofluids/magnetically
controlled
suspensions, Colloids and Surfaces A: Physicochem. Eng.
Aspects 217 (2003) 171_/178, Elsevier Science B.V.
[4]. Luca E., Călugăru Gh. s.a., Industrial Applications of
Ferrofluids, (in Romanian), Bucureşti, Editura Tehnică, 1978.
[5]. Odenbach S., Colloidal Magnetic Fluids: Basics,
Development and Application of Ferrofluid, Springer, 2009.
[6]. Odenbach S., Ferrofluids magnetically controlled
suspensions, Colloids and Surfaces, A: Physicochem. Eng.
Aspects, vol. 217, no. 171/178, Elsevier, 2003.
[7]. Pop L., Odenbach S, s.a., Microstructure and rheology of
ferrofluids, Journal of Magnetism and Magnetic Materials,
vol. 289, pp. 303–306, Elsevier, 2005.
[8]. Meeker D. Finite Element Method Magnetics, Version 4.2,
User’s Manual October 16, 2010.
[9]. Meeker D. Finite Element Method Magnetics:OctaveFEMM,
Version 1.6, User’s Manual October 16, 2010.
[10]. Călăraşu D., Ciobanu B., Fluids Mechanics. Ferrofluids
applications, (in Romanian), Iaşi: Editura Tehnică, Ştiinţifică
şi Didactică CERMI, 2005.
[11]. www.ferrotec.com/index.php?id=audioFluid&vfp_id=49
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