Accepted Manuscript
Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: analytical solution and numerical modeling O. Budenkova, N. Bernabeu, S. Rukolaine, Y. Du Terrail Couvat, A. Gagnoud, R. Tarpagkou, Y. Fautrelle PII: DOI: Reference:
S0307-904X(17)30021-5 10.1016/j.apm.2017.01.016 APM 11513
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
11 May 2016 20 November 2016 5 January 2017
Please cite this article as: O. Budenkova, N. Bernabeu, S. Rukolaine, Y. Du Terrail Couvat, A. Gagnoud, R. Tarpagkou, Y. Fautrelle, Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: analytical solution and numerical modeling, Applied Mathematical Modelling (2017), doi: 10.1016/j.apm.2017.01.016
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Highlights • Current density inside the ellipsoid particle does not depend on the particle
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size • Current density inside the ellipsoid particle depends on ellipse’s eccentricity
• Current density in the ellipsoid depends on its orientation in the thermal field
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• Inside an ellipsoid the vectors of current and thermal gradient may be not parallel
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• The ellipsoid particle is translated by the thermoelectric force
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Thermoelectric problem for an axisymmetric ellipsoid particle in the liquid metal: analytical solution and numerical modeling
a CNRS, SIMAP, F-38000 Grenoble, France Grenoble Alpes, SIMAP, F-38000 Grenoble, France c Ioffe Institute, 194021 St.Petersburg, Russia
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b Univ.
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O. Budenkovaa,∗, N. Bernabeub , S. Rukolainec , Y. Du Terrail Couvatb , A. Gagnouda , R. Tarpagkoub , Y. Fautrelleb
Abstract
A thermo-electric problem is solved analytically for an electrically conducting particle in a form of an ellipsoid of revolution immersed in the liquid metal and subjected to a temperature gradient. It is shown that the density of the thermoelectric current is constant inside the particle and its value depends on
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the eccentricity of the ellipse in the meridian plane of the ellipsoid, but does not depend on the size of the particle. Another parameter which affects the value of
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the thermoelectric current is the orientation of the ellipsoid with respect to the imposed temperature gradient. The vector of the thermoelectric current inside the particle and the vector of the imposed thermal gradient are co-planar, but
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a planar angle between these vectors exist and its value is also a function of the eccentricity of the ellipse and its orientation in a thermal field. Limiting minimal
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and maximal value of the thermoelectric current inside a very elongated particle are found and compared with values obtained in simulations for a dendrite grain. Numercial simulation performed with FEM software for two orientations of an
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elongated ellipsoid with respect to the imposed thermal gradient provided results similar to analytical solutions with the relative error less then 0.1%. Keywords: thermoelectric effects, alloys solidification, curvilinear coordinates, ∗ Corresponding
author Email address:
[email protected] (O. Budenkova)
Preprint submitted to Applied Mathematical Modelling
January 10, 2017
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analytical solution, finite element method
1. Introduction
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A thermo-electric effect in metallic systems is related to the dependence of
the mobility of electrons on temperature and is resulted in the appearance of the
electric current if two metals are brought to the contact subjected to a thermal 5
gradient. In material processing the thermo-electromagnetic phenomenon was discovered in crystal growth experiments where thermal and electrical proper-
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ties of the material varied between the solid and the liquid phase [1]. In these experiments a static magnetic field was applied to damp the natural convection, however, a Lorentz force which appeared in the liquid due to the interaction 10
of the thermoelectric current and the magnetic field caused convection that lead to the deformation of the crystallization front. During the last decades the idea to use a magneto-thermoelectric force during solidification in order
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to affect the solidifying structure was put into practice for metallic alloys [2] and now numerous experimental paper can be found on the subject, see for 15
example [3] and references within. Among them, experiments with in-situ ob-
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servation of the solidification processes via X-rays demonstrated a direct effect of the magneto-thermoelectric force on equiaxed dendrite grains which resulted
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in their horizontal motion [4, 5, 6]. One can cite several publications on the numerical modelling of the thermoelectric effect in two dimensions [7, 8], and in 20
three dimensions [9, 10]. However, simulations of the thermoelectric phenom-
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ena in solidification is still challenging because of inevitable three-dimensional effects, combination of an extremely small scale for the thermoelectric phenom-
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ena and a large spread for convective flows and strong coupling between the fluid flow and the intensity of the thermoelectric current. In most numerical
25
works certain simplifications were made for the models, that, in fact, impedes clear understanding of the phenomena. Moreover, even in the absence of fluid flow there is a lack of theoretical results that makes analysis of complex simulations quite difficult. Analytical solution of a thermoelectric problem obtained
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for the spherical particle in [11] and reassessed recently in [12] was an indis30
pensable step both for comprehension of the phenomenon and development of numerical tools for further modeling. In particular, analytical solution showed
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that the thermoelectric current should be constant inside the solid sphere and directed parallel or anti-parallel to the imposed temperature gradient according to the difference of the thermoelectric power in the solid and the liquid. Fur35
thermore, the density of the electric current was found to be independent on
the size of a sphere and defined only by the thermal and electric properties of the materials. On the other hand, performing 2D simulations for a dendrite
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particle using a multi-grid FEM based software AEQUATIO [13] we found that the thermo-electric current tends to flow along the dendrite branches as shown 40
in Fig. 1b, i.e. its direction deviates from that of the imposed temperature gradient. Furthermore, the maximal value of the density of the thermoelectric
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current exceeded significantly the value estimated for a spherical particle.
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Figure 1: Results of 2D calculations of a thermoelectric current in FEM software AEQUATIO, with properties given in the table 1 and imposed thermal gradient 3000K/m. a)Image of a dendrite extracted from experimental data; b) thermoelectric current flow in the dendrite, vector length is proportional to the density of the current c) distribution of the density of the
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electric current inside the dendrite. Temperature isolines are shown with ∆T = 0.05K.
Consequently, the present work has been done to demonstrate the effect of
the shape of a solid particle on the thermoelectric current in the solid and in the
45
liquid phase and on the resulting thermoelectric force acting on the particle if an external magnetic field is imposed on the system. It must be noted that the 4
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thermoelectric problem for the ellipsoidal particle was solved in [14] based on a method proposed in [15]. However, the attention was given to the magnetic field created by the thermoelectric current while formulation of the results did not allow to draw conclusions made in the present paper. We propose another
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50
way for the solution of the thermoelectric problem accompanied with numerical simulations realized for the ellipsoid particle.
2. Statement of the problem. Transition to the elliptical coordinates In the problem we consider a metallic particle of a form of a prolate spheroid
(ellipsoid of revolution) which is immersed in the electrically conducting infinite
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55
liquid, as shown in Fig. 2. A meridian section of the particle in the plane (x, y) is an ellipse with semi-major and semi-minor axis a and b, respectively, and c is a distance from the center of the ellipse to their focii shown in Fig. 2 with dots.
y
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ey e z ex
b
c
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-c
a
x
z
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G∞
ε
Figure 2: Ellipsoid particle immersed in the liquid and subjected to a thermal gradient G∞ : a
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and b are semi-major and semi-minor axis of the ellipsis in the meridian plane, c is a distance
from the center of the ellipsis to their focii; ε is the angle in y, z plane starting from the y-axis
The eccentricity of the ellipse which is defined as
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e = c/a
(1)
is a measure of its deviation from the circle as well as a deviation of the ellipsoid from a sphere. Any cross-section of the ellipsoid parallel to the (y, z) plane is a
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circle. The thermal and electrical conductivities, κ and σ, respectively, are constant
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inside each media but different for the solid and the liquid and the same is applicable to the thermoelectric powers (Seebeck coefficients) S. The latter
defines the dependence of the electron mobility on temperature. The system is subjected to a temperature gradient G∞ which can be directed arbitrary with 70
respect to the axes of the ellipsoid particle. As it is shown in the present work, the solution for a general orientation of G∞ can be obtained as a combination of k
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the solutions obtained for a temperature gradient directed parallel to the major k
axis of the ellipsoid, i.e. G∞ = ex G∞ (according to notations in Fig. 2) and perpendicular to it G⊥ ∞ , which, due to the axial symmetry of the system can be 75
⊥ chosen directed along the y-axis: G⊥ ∞ = ey G∞ .
2.1. Mathematical statement of the problem
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The system of governing equations includes differential equation for temperature distribution T and continuity of the electric current j as well as relation
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between the density of the electric current and the imposed temperature gradient. These equations can be written for each media i, with i = S for the solid
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and i = L for the liquid [16]:
∇2 Ti = 0
(2)
∇ · ji = 0
(3)
ji = −σi ∇Ui − σi Si ∇Ti
(4)
In (4), U is an electric potential which appears in a media due to the temperature
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gradient. As it was indicated in [16], for further analysis it is convenient to introduce a generalized potential Wi for each media: Wi = Ui + Si Ti
6
(5)
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With Wi , equation (4) and then (3) can be rewritten as (6)
∇2 W i = 0
(7)
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ji = −σi ∇Wi
Boundary conditions which are required to close the system of equations (2), (5),(6) and (7), presume continuity for the temperature T and for the electric 85
potential U at the surface of the particle, as well as for the heat flux and electric current through the solid-liquid interface that gives the following equations:
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TL |w = TS |w (WS − WL )|w = (SS − SL )T |w ∂TS ∂TL = κS κL ∂n w ∂n w ∂WL ∂WS σL = σS ∂n w ∂n w
(8) (9)
(10) (11)
where (∂/∂n) denotes the normal derivative with respect to the surface of the
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particle. At infinite distance from the particle the temperature is defined by the
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imposed gradient while the electric current disappears: ∇TL |∞ = G∞
(12)
jL |∞ = 0
(13)
It should be noted that the statement of the problem is quite similar to
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the one about a flow of an ideal fluid around a body of revolution. The latter
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attracted enormous attention in the beginning of the last century due to the development of aviation and artillery and for some body profiles was solved analytically using the potential of velocity [17, 18]. Consequently, we used the same approach which consisted in transformation of the coordinate into
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elliptical ones and in choosing base functions for a general solution of the partial differential equation in a curvilinear coordinate system. However, the difference between the present problem and the one about the flow is significant since no velocity is developed in the solid body while in the present case the heat
100
flux and the electric current both pass through the particle. Moreover, we are 7
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interested in the electric current inside the solid particle and how it is affected by the shape of the latter. In [14] mentioned above, the thermoelectric problem was solved using parametrisation in a cartesian coordinate system and known
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integral solution for the Laplace equation. 2.2. Elliptical coordinates
For a particle with the shape of a prolate spheroid (ellipsoid of revolution), the stated problem is convenient to solve using ”axisymmetric” confocal elliptical coordinates ξ, η, which are related to the Cartesian one as [17, 18]:
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x = c cosh ξ cos η, 0 ≤ ξ < ∞ y = c sinh ξ sin η cos ε, 0 ≤ η ≤ π z = c sinh ξ sin η sin ε, 0 ≤ η ≤ π
where ε is the angle between the axis y and a chosen meridian plane as shown 110
in Fig. 2. Further, contours ξ = const corresponds to ellipses and contours of
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η = const are hyperbolas perpendicular to ellipses as shown in Fig. 3.
y
η = η4
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η = η6
ξ = ξ4 ξ = ξw b ξ = ξ2 ξ = ξ1 = 0
η = η2 η = η1
a x
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η = η6
η = η3
Figure 3: Confocal elliptical coordinates
With a commonly accepted notation cosh ξ = λ, 1 ≤ λ < ∞ cos η = µ, −1 ≤ µ ≤ 1 8
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obtain: (14) (15)
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x = cλµ p p y = c (λ2 − 1) (1 − µ2 )cosε p p z = c (λ2 − 1) (1 − µ2 )sinε
(16)
One of the major convenience of the elliptic coordinate system is that the surface 115
of the particle corresponds to a fixed value of one of the coordinates: λ = λw ,
and that the latter is related to the eccentricity of the ellipse e in the meridian
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section given by (1) as λw = 1/e
(17)
The stated problem of the thermoelectric current requires solutions of the two Laplace equations, (2) and (7), the latter for a scalar function ϕ in a chosen
(18)
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coordinate system λ, µ, ε becomes [18, chapter VII, p.381-479]: 2 ∂ 1 ∂ϕ ∂ 1 ∂ ϕ 2 2 ∂ϕ (λ − 1) + (1 − µ ) + + =0 ∂λ ∂λ ∂µ ∂µ λ2 − 1 1 − µ2 ∂ε2
Note, that the normal derivative at the surface of a particle in the elliptical
boundary conditions (8)-(11) can be rewritten in the form:
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120
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coordinate corresponds to a derivative over the coordinate λ. Consequently, the
TL,λw = TS,λw
(19)
WS,λw − WL,λw = (SS − SL )TS,λw ∂TL ∂TS κL = κS ∂λ λw ∂λ λw ∂WS ∂WL = σS σL ∂λ λw ∂λ λw
(20) (21) (22)
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Further analysis requires separate consideration for different orientations of the imposed thermal gradient G∞ with respect to the particle. To distinguish solutions, we shall use the sign k for those obtained in the case when the imposed k
gradient is parallel to the major axis of the ellipsoid, i.e. G∞ , and the sign ⊥
125
if G⊥ ∞ is imposed. Then, a general solution for the thermoelectric current will be presented as a combination of two solutions. 9
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3. Case 1: imposed thermal gradient Gk∞ parallel to the major axis of the ellipsoid k
For the thermal gradient G∞ imposed parallel to the x-axis, the solution for
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any field does not depend on ε because of the axial symmetry of the problem. Laplace equation (18) in this case has the form ∂ ∂ ∂ϕ ∂ϕ (λ2 − 1) + (1 − µ2 ) =0 ∂λ ∂λ ∂µ ∂µ
Using variable separation, a general solution for the latter equation is sought in
the form ϕ = L(λ)M (µ) which allows one to split it into two ordinary differentials equations:
∂ 2 ∂L (1 − λ ) + n(n + 1)L = 0 ∂λ ∂λ ∂ ∂M (1 − µ2 ) + n(n + 1)M = 0 ∂µ ∂λ
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130
where n is a positive integer number. These equations are of Legendre type and polynomials of two types satisfy them:
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1) Legendre polynomials of first type Pn (t):
P0 (t) = 1
(23)
P1 (t) = t
(24)
···
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etc, with a recurrence relation
(n + 1)Pn+1 = (2n + 1)tPn (t) − nPn−1 (t)
Note that with t → ∞ these functions grow to infinite.
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2) Legendre polynomials of the second type Qn (t): 1 t+1 ln 2 t−1 1 t+1 Q1 (t) = t ln −1 2 t−1 ··· Q0 (t) =
(25) (26)
etc, with the same recurrence relation. Further solution requires separate consideration for the temperature and electric potential. 10
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k
3.1. Solution for the temperature field, case G∞ Let us present the solution for the temperature field, both for the solid and
140
k k k Ti = Gk∞ x + T˜i = Gk∞ cλµ + T˜i
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the liquid i = S, L, in the form
k where T˜i is a perturbation of the temperature field, which behaves differently in
the liquid and in the solid. Indeed, at the infinite distance form the particle in
k the liquid, according to the condition (12), T˜i should disappear, that means that 145
the Legendre polynomials of the second type should be used for the coordinate λ.
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Moreover, since no additional heat sources (apart from the imposed gradient) exist in space, the temperature perturbation should be proportional to λ−2 .
Accounting for the approximations at λ → ∞ given in the Appendix A, one can see that the series containing Qn (λ) should start with n = 1. This means 150
that temperature distribution in the liquid should be presented in a form: k
∞ X
An · Qn (λ) · Pn (µ)
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TL = Gk∞ cλµ +
n=1
(27)
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For the temperature distribution in the solid particle, on the contrary, positive exponents of λ are required since solution should be bounded at λ = 1, therefore Legendre polynomials of the first type should be used for both coordinates.
155
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Accordingly, the temperature distribution in the solid should be presented in the form:
=
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k TS
Gk∞ cλµ
+
∞ X
n=1
Bn · Pn (λ) · Pn (µ)
(28)
Retaining only the first term of the series (see section 5.2) and using bound-
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ary conditions (19) and (21), obtain the following equations for the unknown coefficients A1 and B1 :
κL
A1 · Q1 (λw ) · P1 (µ) = B1 · P1 (λw ) · P1 (µ) # " # dQ1 dP1 k k G∞ cµ + A1 P1 (µ) = κS G∞ cµ + B1 P1 (µ) dλ λw dλ λw
"
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Recalling that P1 (µ) = µ, P1 (λw ) = λw and dP1 (λ)/dλ = 1 according to (24), one can find coefficients A1 and B1 : " A1 =
B1 =
k G∞ c (κS
k G∞ c (κS
− κL ) κL "
dQ1 dλ
λw − κL ) κL Q1 (λw )
λw
Q1 (λw ) − κS λw
dQ1 dλ
#−1
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λw
− κS
#−1
Taking the derivatives, and inserting A1 and B1 in a general solution, one can
obtain temperature distribution in the liquid and in the solid. These cumbersome relations can be simplified with use of the eccentricity of the ellipse in the
165
notation
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meridian section e and its relation with the coordinate λw (17). Then, with a
kκ = κS /κL
(29)
these solutions are written as follows: k
k
−1
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TS = G∞ cλµ [1 − (1 − kκ )F(e)] 1 λ+1 1 k k k ln − TL = G∞ cλµ + G∞ cλµE(e)(1 − kκ ) 2 λ−1 λ −1
1 1 1+e F(e) = −1 ln −1 e2 2e 1 − e 1 1 −1 E(e) = e e2
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where
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× [1 − (1 − kκ )F(e)]
(30)
(31)
(32) (33)
Solution given by (30)-(33) can be verified by the one obtained for a sphere of
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a radius R0 [11, 12] as it is shown in Appendix B.
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k
3.2. Solution for the generalised electric potential, case G∞ Similar to the temperature field, solution for the generalized potential can
be presented in a form Wi = L(λ)M (µ) where L(λ) and M (µ) are Legendre
polynomials given above. With the same argumentation for the bounded behavior of the potential inside the particle and absence of the electric current at 12
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175
infinite distance from the latter (13), general solutions in two media are sought in the forms:
k
WL =
P∞
n=1
P∞
n=1
Bn · Pn (λ) · Pn (µ) An · Qn (λ) · Pn (µ)
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k
WS =
Retaining the first term (see discussion in section 5.2) and recalling (24), ob-
tain from the boundary conditions (20)-(22) the following expressions for the coefficients A1 and B1 :
#−1 Q1 (λw ) dQ1 σS − σL λw dλ λw λw " #−1 1 Tw Q1 (λw ) dQ1 σS (SL − SS ) σS − σL A1 = µ λw λw dλ λw
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dQ1 dλ
"
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1 Tw B1 = σL (SL − SS ) µ λw
Computing temperature at the surface with required derivatives and using the relation (λw )−1 = e, obtain for the generalized potential in the solid and in the
k
k
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liquid the following solutions:
WS = G∞ cµλ(SL − SS )(F(e) − 1)[1 − (1 − kσ )F(e)]−1 −1
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× [1 − (1 − kκ )F(e)] 1 λ+1 1 (SL − SS ) k k WL = G∞ cµλ ln − E(e) 2 λ−1 λ kσ −1
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×[1 − (1 − kσ )F(e)]−1 [1 − (1 − kκ )F(e)]
(34)
(35)
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where
kσ = σS /σL
(36)
and F(e) and E(e) are given by (32)-(33). Obtained expressions can be verified by transition to a sphere as shown in Appendix B.
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k
3.3. Thermoelectric current in the system, case G∞ Expressions for the thermoelectric current in the solid and in the liquid can
be obtained by taking derivatives of the generalized potentials given by (34) and (35), respectively. Furthermore, recalling that cµλ = x, one can easily see that 13
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190
the generalized potential in the solid phase depends only on the x coordinate. Therefore, the thermoelectric current in the solid phase is parallel to the major axis of the ellipsoid while its value depends on the eccentricity of the ellipse in
k
k
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the meridian section: jS (e) = −ex G∞ σS (SL − SS )(F(e) − 1)[1 − (1 − kσ )F(e)]−1 −1
× [1 − (1 − kκ )F(e)]
(37)
In the simplest case if kκ = 1, the temperature field in space is not disturbed 195
by the particle since its thermal conductivity is equal to that of the liquid, then
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if kσ = 1 one can get rid of the both denominators in (37). However, generally kκ > 1 and kσ 6= 1 and for further calculations some values should be assigned
to these properties of material since they are “entangled” in the formulae. In the present work these values were taken similar to the ones in [12] and are 200
given in the Table 1. In all calculations the imposed gradients were taken equal k
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to 3000K/m, i.e. in the present case G∞ = 3000K/m.
Table 1: Numerical values of physical properties used in calculations
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Physical properties
Thermo-electric constant S Electrical conductivity σ
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Thermal conductivity κ
value
unit
liquid
solid
−3.8 × 10−6
−5.4 × 10−6
95
150
4 × 106
1 × 107
V m−1 Ω−1 m− 1 W m−1 K−1
k
The dependence of jS (e) on the eccentricity of the ellipse (37) is presented in k
Fig. 4. A minimal value of the density of the electric current jS (e) corresponds
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k
to e = 0, i.e. to the case of a spherical particle, further jS (e) increases with
205
elongation of the ellipsoid. Whether the thermoelectric current is parallel or
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anti-parallel to the imposed thermal gradient depends only on the sign of the difference of Seebeck coefficients in the solid and in the liquid (SL − SS ). This difference is positive in the present case, then F(e) < 1 for 0 ≤ e ≤ 1, therefore, k
k
for the given properties jS is parallel to the G∞ .
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The remarkable finding is that the value of the thermoelectric current remains limited for the elongated ellipsoids with eccentricity close to 1. In this 14
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55000 50000 45000 40000 35000 30000 25000 20000
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jS , mA2
kσ = 1 k =1 0
0.2
0.4
0.6
0.8
1
e
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Figure 4: Dependence of the value of the thermoelectric current in an ellipsoid particle on
the eccentricity of the ellipse in its meridional plane if the thermal gradient is parallel to the k
k
major axis of the ellipse, G∞ = ex G∞ : solid line is calculated for the properties given in the table 1, dashed line is for kκ = 1 while other properties are kept and dotted-dashed line is for kσ = 1, other properties are kept
k
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case F(e)|e→1 → 0, consequently:
jS |e→1 → ex Gk∞ σS (SL − SS )
(38)
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and for the given properties jk (e → 1) = 55 · 103 A/m2 . In the liquid surrounding the ellipsoid particle the thermoelectric current has 215
two components which can be obtained readily from the generalized potential k
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WL using differentiating rules for the orthogonal curvilinear coordinate system
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[18, chapter VII, p.381-479]: k jL,λ
k jL,µ
= −σL c = −σL c
s
s
k
λ2 − 1 ∂WL λ2 − µ2 ∂λ
k
1 − µ2 ∂WL λ2 − µ2 ∂µ
(39) (40)
For the sake of brevity these relations are omitted here but the vector field of jL is presented and discussed in the section 6.
15
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220
4. Case 2: imposed thermal gradient G⊥ ∞ perpendicular to the major axis of the ellipsoid If the imposed thermal gradient is directed perpendicular to the major axis
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of the ellipsoid, then without the loss of generality, the y-axis can be chosen in
a way to keep the vector G⊥ ∞ in the plane (x, y). Then a dependence of the 225
solution on the angle ε should be kept into account in the (18), where ε is the
angle taken in the plane (x, z) between the axis y and other directions as shown in Fig. 2. It is appropriate to look for a general solution of (18) in the form:
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ϕ = N (λ, µ)E(ε) that gives two equations after separation of variables:
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d2 E(ε) + l2 E(ε) = 0 dε2 ∂ ∂N (λ, µ) ∂ 2 2 ∂N (λ, µ) (λ − 1) + (1 − µ ) ∂λ ∂λ ∂µ ∂µ 2 2 λ −µ N (λ, µ) = 0 −l2 2 (λ − 1)(1 − µ2 )
where l is an arbitrary number which can be considered as a positive integer.
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The first of the two equations has a general solution in the form E(ε) = A cos lε + B sin lε
For the second equation we can put N = L(λ)M (µ) and separate variables that
PT
gives other two ordinary differential equations: ∂ l2 2 ∂L (1 − λ ) + n(n + 1) − L=0 ∂λ ∂λ 1 − λ2 ∂ ∂M l2 (1 − µ2 ) + n(n + 1) − M =0 ∂µ ∂µ 1 − µ2
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230
Solutions for these two equations are given via associated Legendre functions of
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two types: Pnl (x) and Qln (x), where n is a degree and l is the order. For the
integer l, these functions are related to the Legendre polynomials introduced above by (23)-(26): dl Pn (t) dtl l d Q n (t) Qln (t) = (t2 − 1)l/2 dtl Pnl (t) = (1 − t2 )l/2
16
(41) (42)
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Further solutions for the temperature field and for the generalized potential are presented separately.
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4.1. Solution for the temperature field, case G⊥ ∞ Similar to the previous case, we shall present a general solution for the temperature field both in the liquid and in the solid (i = L, S) in a form Ti⊥ = 240
˜ G⊥ ∞ y + Ti
⊥
⊥ where T˜i is a perturbation which is different for the solid and the
liquid. Decrease of T˜L at infinite because of condition (12) and boundedness of
the solution T˜S in the center of the particle impose different combinations of
TL⊥ = G⊥ ∞y + TS⊥ = G⊥ ∞y +
n=1 ∞ P
n=1
Qln (λ)Pnl (µ)(AL,nl cos lε + BL,nl sin lε) Pnl (λ)Pnl (µ)(AS,nl cos lε + BS,nl sin lε)
p √ 2 2 Note also that using the eq.(15) we can present G⊥ ∞ y = G∞ c λ − 1 1 − µ cos ε.
Further, let us take solutions with l = 1 which contain only cos ε, and let
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245
∞ P
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Legendre functions for the solutions in the solid and in the liquid:
AL,n1 = cG⊥ ∞ CL,n and AS,n1 = cG∞ CS,n where coefficients Ci,n should be defined. Also, recall definitions of Legendre’s associated functions (41)-(42) that
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gives:
dQn (λ) dPn (µ) +1 dλ dµ n=1 ∞ p √ P P dP (λ) n (µ) n 2 − 1 1 − µ2 cos ε = G⊥ c λ + 1 C S,n ∞ dλ dµ n=1
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p √ 2 2 TL⊥ = G⊥ ∞ c λ − 1 1 − µ cos ε TS⊥
∞ P
CL,n
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Holding only the first term of both series and recalling definitions of polynomials P1 (24), obtain:
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250
p √ dQ1 (λ) 2 − 1 1 − µ2 cos ε C TL⊥ = G⊥ c λ + 1 L,1 ∞ dλ p √ ⊥ ⊥ 2 2 TS = G∞ c λ − 1 1 − µ cos ε · D
here D = CS,1 + 1 is a new constant to be defined. Using boundary conditions at the surface of the particle (19)-(21), obtain two equations for the unknown
17
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coefficients:
κL λw CL,1
dQ1 dλ
λw
dQ1 dλ #
+1=D λw
+ 1 + κL (λ2w − 1)CL,1 = κS λw D
d2 Q1 d2 λ
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CL,1 "
λw
Taking derivatives, solve the system of equations for the coefficients CL,1 and 255
D and obtain solutions for the temperature distribution in the liquid and in the solid which can be expressed via the eccentricity of the ellipse:
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TL⊥
p √ −1 2 2 TS⊥ = 2G⊥ (43) ∞ c λ − 1 1 − µ cos ε [2 − (1 − kκ )(1 − F(e))] p p √ √ 2 2 2 2 = G⊥ ∞ c λ − 1 1 − µ cos ε − G∞ c λ − 1 1 − µ cos ε(1 − kκ )E(e) 1 λ+1 λ −1 × ln − [2 − (1 − kκ )(1 − F(e))] (44) 2 λ − 1 (λ2 − 1)
where F(e) and E(e) are defined by the (32) and (33), respectively and kκ is
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a ratio (29). As in previous case, these solutions can be verified by the ones presented in [11, 12] for a sphere taking the series expansion for e → 0 as it is demonstrated in the Appendix C
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260
4.2. Solution for the generalised electric potential, case G⊥ ∞
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Solution for the generalized electric potential can be found in a similar way. Consequently, a general solution for the potential in a phase i = S, L, will take
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the form:
Wi⊥ = Ni (λ, µ)E(ε)
Further, similar to temperature, let Ni (λ, µ) = Li (λ)Mi (µ) where Li and Mi are
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associated Legendre functions defined by (41)-(42). Again using argumentation for solution boundedness in the center of the particle and disappearance of the
265
potential perturbation at infinity, assign Legendre function of the first type Qln (λ) to the liquid and of the second type Pnl (λ) to the solid. Taking the part of the solution which contains only cos ε and then retaining only the first term
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of the series, obtain the following expressions for the generalized potential in the solid and the liquid phases:
WS⊥ = BP11 (λ)P11 (µ) cos ε 270
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WL⊥ = AQ11 (λ)P11 (µ) cos ε
which, similarly to the consideration in the previous section, are transformed into
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p √ dQ1 cos ε WL⊥ = Ac λ2 − 1 1 − µ2 dλ p √ WS⊥ = Bc λ2 − 1 1 − µ2 cos ε
With coefficients A, B to be found from the boundary conditions (20)-(22) which are written as follows:
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" # p p dQ 1 − B = (SL − SS ) T |λw c λ2w − 1 1 − µ2 cos ε A dλ λw " 2 # p d Q1 λw dQ1 λw σL A p + λ2w − 1 = σS B p 2 2 dλ dλ λw − 1 λ2w − 1 λw λw
Finally, obtain the solutions:
−1
−1
× [2 − (1 − kσ )(1 − F(e))] [2 − (1 − kκ )(1 − F(e))] (45) p √ λ + 1 λ 1 2 2 ln − = 2G⊥ ∞ (SS − SL )c λ − 1 1 − µ cos εE(e) 2 λ − 1 λ2 − 1
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WL⊥
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p √ 2 2 WS⊥ = 2G⊥ ∞ (SS − SL )c λ − 1 1 − µ cos ε [1 + F(e)]
−1
−1
[2 − (1 − kκ )(1 − F(e))]
(46)
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×kσ−1 [2 − (1 − kσ )(1 − F(e))]
275
with F(e) and E(e) given by (32) and (33) and ratios (29) and (36) for kκ and kσ . Solutions presented in [11, 12] for a sphere can be obtained from (45)-(46)
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with transition e → 0 as it is demonstrated in Appendix C. 4.3. Thermoelectric current in the system, case G⊥ ∞
280
k
It can be noted that, similar to the previously considered case of the G∞ , the
potential in the solid phase depends only on a combination of the three elliptical coordinates which corresponds to the coordinate y according to (15). Taking 19
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this into account, the density of the thermoelectric current can be obtained by taking a partial derivative of WS⊥ on y in the Cartesian system which simply gives: −1
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⊥ j⊥ S (e) = −ey 2G∞ σS (SS − SL ) [1 + F(e)] [2 − (1 − kσ )(1 − F(e))]
−1
× [2 − (1 − kκ )(1 − F(e))] 285
(47)
In Fig. 5 the dependence of the thermoelectric current j⊥ S (e) on the eccentric-
ity of the ellipse is presented for the physical properties given in the Table 1 and G⊥ ∞ = 3000K/m. In this case the value of the thermoelectric current is
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maximal for a spherical particle and is diminished with ellipsoid flattening. The interesting thing is that for a ”thin” particle, i.e. when e → 1, a finite thermo290
electric current flows through the particle and its value remains dependent on the thermal and electrical conductivities of both media contrary to the previous case:
−1
jS⊥ , mA2
(48)
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35000
−1
[1 + kκ ]
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⊥ j⊥ S (e)|e→1 = −ey 2G∞ σS (SS − SL ) [1 + kσ ]
kσ = 1 k =1
30000
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25000 20000 15000
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10000
0
0.2
0.4
0.6
0.8
1
e
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Figure 5: Dependence of the value of the thermoelectric current in an ellipsoid particle on the eccentricity of the ellipse if the imposed thermal gradient is perpendicular to the major axis ⊥ of the ellipsoid, G⊥ ∞ = ey G∞ : solid line is calculated for the properties given in the table 1,
dashed line is for kκ = 1 while other properties are kept and dotted-dashed line is for kσ = 1, other properties are kept
The components of the thermoelectric current in the liquid can be found 20
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readily by taking derivatives of WL⊥ according to the rules related to the orthogonal curvilinear system [18, chapter VII, p.381-479]: s λ2 − 1 ∂WL⊥ ⊥ jL,λ = −σL c λ2 − µ2 ∂λ s 1 − µ2 ∂WL⊥ ⊥ jL,µ = −σL c λ2 − µ2 ∂µ s 1 − µ2 ∂WL⊥ ⊥ jL,ε = −σL c λ2 − µ2 ∂ε
(49)
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295
(50)
(51)
The final formulas for the thermoelectric current in the liquid are not given here
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for the sake of brevity but discussion is provided in section 6.
5. General case: arbitrary orientation of the ellipsoid with respect to thermal gradient G∞
In a general case the orientation of the ellipsoid with respect to the direction
300
to the imposed thermal gradient can be arbitrary. Because of axial symmetry
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of the particle, the orientation of the x and y axis can be chosen in a way to keep the vector of the thermal gradient G∞ in the plane (x, y). Then the perpendicular to the major axis of the ellipsoid: G∞ = Gk∞ + G⊥ ∞
(52)
Gk∞ = ex G∞ cos α
(53)
G⊥ ∞ = ey G∞ sin α
(54)
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305
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vector of the thermal gradient can be presented as a sum of vectors parallel and
where α is the angle of the vector G∞ with respect to the major axis of ellipsoid,
i.e., the x axis in the plane (x, y). Consequently, due to the linearity of the
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differential equations, the resulting density of the thermoelectric current will be also presented as the sum of the two solutions:
310
j(e) = jk (e) + j⊥ (e)
(55) k
where jk (e) and j⊥ (e) are obtained for the values of the thermal gradients G∞ and G⊥ ∞ , given by (53) and (54), respectively. This can be done easily for 21
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the thermoelectric current in the particle using equations (37) and (47). Note that since the two components of the thermoelectric current inside the particle behave differently with respect to the eccentricity of the ellipse, the direction of the resulting vector j(e)S also varies with elongation of the ellipsoid as it is shown in Fig. 6 for the angle β calculated as |j⊥ S (e)|
β = arctan
k
|jS (e)|
!
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315
(56)
For a small value of e, when the ellipsoid does not differ too much from a sphere, the thermoelectric current follows the direction of the thermal gradient. With
320
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ellipsoid deformation, the component of the current parallel to the major axis k
of the ellipsoid jS (e) increases while the component j⊥ S (e) decreases and the resulting vector jS (e) tends to be aligned with the major axis. This tendency exists even if the thermal conductivities of the solid and the liquid are similar, i.e. if the temperature field is not perturbed by the presence of the particle
325
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(dashed lines in Fig. 6) and it becomes more obvious for kκ 6= 1 since in this case the direction of the vector of the thermal gradient in the solid also differs from the direction of the imposed thermal gradient G∞ as it follows from (30)
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and (43).
Variation of the module of the resulting thermoelectric current jS (e) with ellipsoid elongation is presented in Fig. 7 for different angles α of the vector of the imposed thermal gradient G∞ with respect to the major axis of the
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330
ellipsoid. For e ≤ 0.3 the module of the thermoelectric current remains almost
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constant with variation of the direction of G∞ . For more elongated ellipsoids
the thermoelectric current decreases if α increases because the input of a larger k
component jS (e) decreases proportionally to the cos α. One can observe a nonmonotone behavior of |jS (e)| for α = 75◦ which is related to the competition
AC 335
between the decrease of the component j⊥ S (e) and increases of the component k
jS (e).
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β, °
80 70 60 50 40 30 20 10
α = 75 ◦ α = 60 ◦
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α = 45 ◦ α = 30 ◦ α = 15 ◦
0.2
0.4
0.6
0.8
1
e
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Figure 6: Angle of the vectors of the thermoelectric current (56) in a meridian plane of the
particle with respect to its major axis: variation with elongation of the ellipsoid for different orientations of the imposed thermal gradient G∞ equal to 3000K/m and imposed with the angle α according to eq.(53)-(54). Data for solid lines are calculated with properties given in the Table 1, dashed lines are for kκ = 1
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kjk, A/m 2 60000
50000
α = 30 ◦
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40000 30000 20000
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10000
0
0.2
α = 15 ◦
α = 45 ◦ α = 60 ◦ α = 75 ◦ 0.4
0.6
0.8
1
e
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Figure 7: Variation of the module of the thermoelectric current in the particle with elongation of the ellipsoid for different orientations of the imposed thermal gradient G∞ , given with the
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angle α according to eq.(53)-(54). Data are calculated with properties given in the Table 1
5.1. Magneto-thermoelectric force acting on the particle
340
If a uniform static magnetic field B is imposed over the system, as it is
usually done in the experiments on solidification, a Lorentz force, referred also
23
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as a magneto-thermoelectric force, acting on the particle appears: fTEM = j × B = jk (e) × B + j⊥ (e) × B
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where the vectors jk (e), j⊥ (e) and G∞ are co-planar. Consequently, if the vector
of magnetic induction B and the vector of thermal gradient are co-planar, the resulting force does not exist. The highest value of the fTEM can be obtained if 345
B is perpendicular to the plane which contains G∞ , i.e. according to the choice of the axis above, when B = ez B. Then, in a general case the value of the fTEM
will depend on the eccentricity of the ellipse e and will be proportional to the |jS |
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presented in Fig. 7. Furthermore, since the thermoelectric current tends to be
parallel to the major axis of the ellipsoid, a particle with e 6= 0 will experience the 350
force acting on it in two directions. In particular, an ellipsoid particle will tend to be moved in a direction perpendicular its major axis. Without convective flow taken into account, the ellipsoid particle will experience only translation since the force is uniform in this case. However, a dendrite grain will be subjected
355
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to a rotation because of different orientation of dendrite arms with respect to the imposed thermal gradient, and, consequently, different intensity of the jS
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in them. Also, convective flow in the liquid may create pressure and stresses which vary along the particle that can also lead to the rotation of the latter. The magneto-thermoelectric force acting in the liquid is discussed below in
5.2. About accounting for the other terms in the series
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360
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section 6.
In all solutions obtained above only the first term of the infinite series was
retained. Using for the illustration the temperature distribution in the case of k
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G∞ let us show that other terms of the series should be equal to zero. For the series given by (27) and (28) with the equality µ = P1 (µ) the boundary
365
condition (19) and (21) imply that P∞ A1 Q1 (λw )P1 (µ) + n=2 An Qn (λw )Pn (µ) = P∞ B1 P1 + n=2 Bn Pn (λw )Pn (µ) 24
(57)
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and # dQn κL Pn (µ) = + n=1 An dλ λw # " P∞ dPn k Pn (µ) κS G∞ cP1 (µ) + n=1 Bn dλ λw
P∞
k G∞ cP1 (µ)
(58)
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"
The Legendre polynomials Pn are orthogonal, hence, linearly independent. Therefore, the factors of Pn (µ) in the (57)-(58) must be equal. As a result we have the system of equations for An and Bn
and κL An
dQn dλ
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κL
An Qn (λw ) = Bn Pn (λw ), n = 1, 2, 3, . . . , " # # dP1 dQ1 k k = κS G∞ c + B1 G∞ c + A1 dλ λw dλ λw
"
= κS Bn
λw
dPn dλ
,
n = 2, 3, . . .
λw
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The inequalities Pn (λ) > 0, (dPn /dλ) > 0, Qn (λ) > 0 and dQn /(dλ) < 0, where
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n = 1, 2, 3, . . . and λ > 1, lead to the inequalities Qn (λw ) Pn (λw ) det > 0, dPn dQn κL κS dλ λw dλ λw
n = 1, 2, 3, . . .
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Therefore An = 0 and Bn = 0, n = 2, 3, . . ., while the coefficients A1 and B1 satisfy the system of equations (57) and (58). In the same manner it can be shown for other series that the terms with indices higher than 1 should be equal to zero. Also, a reader can refer to [17] where a rigorous analysis for all terms
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370
in series was made in another way.
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6. Comparison of the analytical solution with numerical results To demonstrate the main features found analytically and to visualize the
thermoelectric current in the liquid, three-dimensional simulations with FEM
375
software Rheolef [19] were performed for the ellipsoid with e = 0.9, properties and thermal gradient are the same as in the Table 1. To diminish the effect of 25
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the walls on the solution, the size of the calculation domain was taken 25 times larger than the semi-minor axis of the particle and adaptive meshes were used to calculate properly the thermoelectric current inside the particle ( Fig. 8). The major axis of the ellipsoid was kept vertical while the orientation ot the
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380
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thermal gradient varied. Results of calculations for two orientations of the
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Figure 8: Adaptive meshes used in simulations: a)view of a mesh in the calculation domain; b) a mesh inside the ellipsoid particle presented in the meridian plane of the ellipsoid
ED
thermal gradients are presented in Fig. 9 for a meridian plane of the ellipsoid which contains the vector G∞ . For the first case, the angle between the vector G∞ and the major axis of the ellipsoid is α = 45◦ , calculated components of the k
thermoelectric current are jS ≈ 24426 A/m2 and jS⊥ ≈ 10753 A/m2 which gives
PT
385
the module of the thermoelectric current |j|S ≈ 26688 A/m2 and inclinations of
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the vector of jS with respect to the major axis of the ellipsoid β ≈ 23.76◦ . The
module of the thermoelectric current and the angle predicted by the theory are |j|S,theory ≈ 26698A/m2 and ≈ 23.735◦ , respectively (see Fig. 6 and Fig. 7).
AC
390
For the second case, the angle between the vector G∞ and the major axis
of the ellipsoid is α = 75◦ , calculated components of the thermoelectric current k
are jS ≈ 8942 A/m2 and jS⊥ ≈ 4690 A/m2 which gives the module of the
thermoelectric current |j|S ≈ 17197 A/m2 and inclinations of the vector of jS
with respect to the major axis of the ellipsoid β ≈ 58.67◦ . The module of the
395
thermoelectric current and the angle predicted by the theory are |j|S,theory = 26
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17190A/m2 and ≈ 58.64◦ , respectively.
Figure 9: Vectors of the thermoelectric current and temperature isolines with ∆T = 0.031K in a meridian plane of the ellipsoid with e = 0.9, a black solid line indicates the direction of
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G∞ a dashed line indicates the direction of the thermal gradient in the solid ; a) α = 45◦ , |j|S ≈ 26688 vs |j|S,theory ≈ 26698A/m2 , β ≈ 23.76◦ vs βtheory ≈ 23.735◦ ; b) α = 75◦ ,
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|j|S ≈ 17197 vs |j|S,theory = 17190A/m2 , β ≈ 58.9◦ vs βtheory ≈ 58.642◦
Since the thermoelectric current is constant inside the particle and since it disappears at the large distance from the latter, the current lines should form
400
PT
a loop in the liquid to be closed. This can be demonstrated analytically using expression for the generalised potential (35) or (46) and relations (39)-(40) or (49)-(51), respectively, for the components of the thermoelectric current in the
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liquid. Numerical simulations allowed us to visualize these loops and to observe their displacement toward the poles of the ellipsoid or its extremities depending
AC
on the orientation of the particle with respect to the imposed thermal gradient,
405
see Fig. 9. Since jL is not uniform in the liquid and forms the loops, this means that
even for a uniform magnetic field imposed on the system the rotational part of the Lorentz force should differ from zero and therefore leads to the convective flow around the particle. Analysis of the flow and its back effect on the 27
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410
thermoelectric current due to the appearance of the eddy current presents an interesting problem but is out of scope of the present paper.
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7. Conclusion In the work analytical solutions were obtained for a thermoelectric problem
formulated for an electrically conducting particle of the shape of ellipsoid of 415
revolution immersed in the infinite domain filled with the liquid, the latter is subjected to a thermal gradient. Derived equations complement the theoretical
solution for a spherical particle [11, 12] and explains the effect of the shape
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and orientation of the particle on the thermoelectric current which arises in the
system. It must be noted that solution for the ellipsoid under thermal gradient 420
imposed parallel or perpendicular to the major axis of the ellipsoid was obtained in [14] and one can recognize the same behaviour for the thermoelectric current in the solid characterized by the only one component (see eqs. (2.43),(2.45)
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and (2.47) in [14]). Also, coefficients given by eqs.(2.20) looks quite similar to the ones obtained in the present work. However, presentation of the coefficient 425
A0 [14, p.34, eq.(2.35a)] with λ = 0 contains tan−1 whereas coefficients F(e)
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(32) and E(e) (33) used in the present work are expressed via ln that impedes comparison of the two solutions.
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Theoretical analysis performed in the work allowed us to explain the results obtained in the simulation of the thermoelectric phenomena in a dendrite par430
ticle presented in Fig. 1. Indeed, for a dendrite trunk which is elongated and
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whose direction deviates weakly from that of the thermal gradient, the density of the thermoelectric current is not far from the value predicted by (38). For a dendrite arm which is more parallel to temperature isolines, the density of the
AC
thermoelectric current is lower and tends to be close to the value given by (48).
435
Of course, results of numerical calculations for the particles of complex
shapes can differ from the analytical solutions presented here. However, the latter provide a good ground for the analysis of experimental and numerical results in solidification processes. Obtained solution can serve as a numerical
28
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benchmark to test a developed software.
440
8. Acknowledgement
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R.Tarpagkou acknowledge the financial support of LABEX Tec21 (Investissements d’Avenir - grant agreement ANR-11-LABX-0030). The authors are grateful to Dr K.Vankov for the help with SageCloudMath [20] which was used for
the calculation and presentation of the analytical solutions. We thank reviewers 445
for precious comments and indications for the bibliography.
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Appendix A. Legendre polynomials
First, note that Legendre polynomials of the second type Qi (t) can be expanded in series for t → ∞ that gives:
M
1 1 1 Q0 (t)|t→∞ = + 2 + ... = O t 3t t 1 1 1 + 2 + ... − 1 = O 2 Q1 (t)|t→∞ = t t 3t t
Therefore, if solution for a perturbation of the temperature field T˜L is chosen proportional to Q0 , the corresponding density of its heat flux is proportional
ED
450
to t−2 . Let t → ∞ and account for the fact that the surface of a hypothetical sphere surrounding the particle is growing proportional to t2 . This means that
PT
the total heat flux through this surface due to perturbation of the temperature field by the presence of the particle remains constant even at infinite distance 455
from the particle. This contradicts to the condition which we imposed at the
CE
infinite distance. Therefore, the solution both for the T˜L and, for the same
reason, for the generalized potential WL should start with Q1 (t).
AC
For solutions inside the solid particle, T˜S and WS , Legendre polynomials of
the first type should be used since Qi (t) diverge with t → 1
460
Appendix B. Transition to a sphere, case Gk∞ With transition to a sphere of a radius R0 , the distance to the ellipsoid focii goes to zero as well as the eccentricity of the ellipse: c → 0, e → 0, furthermore, 29
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cλ → R and µ → cos θ where R and θ are spherical coordinates. Also, note that e = c/R0 . Further, recalling that in the liquid λ > λw > 1, perform series expansion for the functions depending on e and on λ: 1+e 1 ln = 2 e + e3 + · · · 1−e 3 λ+1 1 + 1/λ 1 1 ln = ln =2 + 3 + ··· λ−1 1 − 1/λ λ 3λ λ 1 1 = 1 + 2 + ··· λ2 − 1 λ λ
(B.1)
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465
(B.2)
(B.3)
at e → 0: F(e)|e→0 =
1 1 1 3 1 = −1 2 e+ e −1 e2 2e 3 3 e→0 1 E(e)|e→0 = 3 e
Then = k
(B.5)
−1 1 3 k 1 − (1 − kκ ) = G∞ R cos θ 3 2 + kκ 1 3(1 − kκ ) k = G∞ R cos θ 1 + 3 3 = 3e λ 2 + kκ R3 1 − kκ k G∞ cos θ R + 02 R 2 + kκ
k G∞ R cos θ
ED
TL |e→0
(B.4)
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k TS |e→0
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Accounting for these expansions, obtain the limit for the function F(e) and E(e)
470
PT
that corresponds to the solution for a spherical particle of a radius R0 [12]. For the potential: k
k
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WS = G∞ cµλ(SL − SS )(1/3 − 1)[(kσ − 1)(1/3 − 1) + kσ ]−1 3 k 2(SS − SL ) −1 R cos θ × [1 − (1 − kκ )1/3] = G∞ 2 + kσ 2 + kκ
AC
and
k k WL |e→0 = G∞ cµλ k
G∞
1 3(SL − SS ) 3 3λ3 e3 kσ (2 + kσ ) 2 + kκ
(SL − SS ) 3 R03 cos θ kσ (2 + kσ ) 2 + kκ R2
30
e→0
=
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Appendix C. Transition to a sphere, case G⊥ ∞ Using series expansions presented above in Appendix B, obtain for the
TL⊥ |e→0
=
G⊥ ∞ R cos θ
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475
temperature in the solid and in the liquid: −1 1 3G∞ R TS⊥ |e→0 = 2G⊥ R cos θ 2 − (1 − k )(1 − ) = cos θ κ ∞ 3 2 + kκ
1 1 3 1 − kκ ⊥ − 3 = − G∞ R cos θ e3 3λ3 λ 4 + 2kκ e→0 1 − kκ R03 G⊥ cos θ ∞ R+ 2 + kκ R 2
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In the same manner, obtain approximation for a sphere for the potential in the solid and in the liquid:
1 3 3 WS⊥ |e→0 = G⊥ (S − S )R cos θ 1 + = S L ∞ 3 4 + 2kσ 2 + kκ
3 2G⊥ ∞ (SS − SL ) R cos θ 2 + kσ 2 + kκ
3 3 = kσ (4 + 2kσ ) 2 + kκ
G⊥ ∞ (SL
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×
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WL⊥ |e→0 = G⊥ ∞ (SS − SL )R cos θ
1 e3
1 1 − 3 3 3λ λ
− SS ) 3 kσ (2 + kσ ) 2 + kκ
R03 R2
e→0
cos θ
Four presented expression corresponds to the solutions obtained for a spherical particle of a radius R0 [12].
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480
References
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[1] A. Mikelson, Y. K. Karklin, Control of crystallization processes by means of magnetic fields, Journal of Crystal Growth 52 (1981) 524–529.
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[2] X. Li, Solidification en pr´esence de champs magn´etiques intenses, Ph.D.
485
thesis, Grenoble, INPG (2007).
[3] X. Li, A. Gagnoud, Y. Fautrelle, R. Moreau, D. Du, Z. Ren, X. Lu, Effect of a transverse magnetic field on solidification structures in unmodified and Srmodified Al-7wt%Si alloys during directional solidification, Metallurgical and Materials Transactions A 47A (2016) 1981–1214. 31
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490
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