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Phenomena in Conductors Having Temperature Dependent Electrical and Thermal Conductivities
G. Fournet
To cite this version:
G. Fournet. Phenomena in Conductors Having Temperature Dependent Electrical and Thermal Con- ductivities. Journal de Physique III, EDP Sciences, 1997, 7 (10), pp.2003-2029. �10.1051/jp3:1997239�.
�jpa-00249698�
Overview Article
Phenomena in Conductors Having Temperature Dependent
Electrical and Thermal Conductivities
G. Foumet(*)
Laboratoire de G4nie (lectrique de Paris (**), (cafe Sup4rieure d'(lectricit4, UniversitAs de Paris VI et XI, Plateau de Moulon, 91192 Gif-sur-Yvette cedex, France
(Received 18 October1996, revised and accepted 18 june 1997)
PACS.44.10.+I Heat conduction (models, phenomenological description)
PACS.84.32 Ff Conductors, resistors (including, thermistors, varistors and photoresistors)
PACS 84.32 Dd Connectors, relays and switches
Abstract. For a conductor of any shape, carrying a current of constant intensity I, and char- acterized by known thermal and electrical conductivities [I(T) and +f(T)], general expressions
are presented which allow one to determine the temperature and electrical potential distribu- tions [T(r) and V(r)]. These expressions involve the conductor's potential %(r) in isothermal regime as a parameter; this potential can be attained by calculation or by measurement when I tends towards zero. The general expressions are explicited for the particular case where I(T)
and T+f(T) are constants.
Rdsumd. Les rApartitions des tempAratures et du poteutiel Alectrique [T(r) et V(r)] au seiu d'uu couducteur de forme quelconque, caractArisA par des couductivitAs thermique et 41ectrique [1(T) et +f(T)] et parcouru par uu courant d'inteusit4 1coustaute, peuveut Atre d4termin4es par des expressions g4n4rales Ces expressions ont pour parambtre le potentiel %(r) du conducteur
en r4gime isotherme; ce poteutiel peut Atre atteiut par le calcul ou par des mesures daus le cas oh l'iutensitd I tend vers z4ro. Les expressions g6udrale sont explicitdes darts le cas oh I(T) et T+f(T)
sont constants
1. Introduction
Whenever one is taking into account -f(T) and I(T) variations of the materials' electrical and thermal conductivities, the determination of the spatial temperature and potential variations
T(r) and V(r), respectively, for a conductor carrying a steady-state current I of constant
intensity constitutes a tricky problem. For the case where the temperature presents a maximum value TM between the extremities of the conductor considered, the problem has already been solved for a long time [1,2j. Later, the contributions by Greenwood and Williamson [3], as
(* e-mail: mondesir@lgep.supelec.fr (**) CNRS D 0127
@ Les (ditious de Physique 1997
well as the book of R. and E. Holm [4] are of particular importance. Also notice a more recent book [5] and especially the paper of Fechant [6]
General analysis of this type of problem (allowing the existence of TM inside the conductor
or not undertaken by us involves an approach at variance with that used by them and leads to the same conclusions. In addition, we were able to attain some new results too. For a system
characterized by the temperatures Ti and T2 Prescribed at the extremities of a homogeneous conductor, as well as by the functions -f(T) and I(T), we have shown the primordial r61e of K
= -f(To)Ro, where Ro is the resistance of the conductor measured in T(r) m To isotherm
regime under very small currents-
independently of the shape of the conductor, the potential difference (Vi V2) and the possible maximal temperature TM depend on the product KI only;
independently of the shape of the conductor, the T(r) and V(r) variations are obtained
by means of universal T(s, KI) and V(s, KI) functions, the shape influencing the sir)
relation only
In this paper are discussed successively: the presentation ofthe basic equations (Sect. 2), the
study of the conditions leading to the coincidence ofthe isothermal and equipotential surfaces,
as well as the definition of these surfaces Sect. 3), the corresponding general expressions of
T(r) and V(r) (Sect. 4), and their application to the particular cases where I(T) a lo and
T-f(T) W C (Sect. 5).
2. Presentation
2.I. DEFINITION OF THE PROBLEM. Our aim is to determine, in (filfit
= 0j steady state
regime, the behavior of a uniform conductor introduced (between the surfaces Si and 52) into
an external electrical circuit carrying a current of intensity (with reference to the direction I
to 2) 1= Ii2. We assume that:
1) on each surface Si and 52 the potential is uniform:
V(r)
= Vi, V r E Si V(r)
= V2, V r E S21 (1)
2) no electrical source is present in the conductor between the surfaces Si and 52, the Thomson effect being neglected;
3) the lateral surface Slat of the conductor (it means the total surface to the exclusion of Si and 52) is electrically insulated; the normal component of the electric field is so zero in each point:
(grad V n)s,a~ % 0. 12)
We intend to study the relations between the current density J and the temperature T distri- bution, assuming that
J(r) = ~t(T(r))E(r) (3)
the thermal flux p being governed by P(r)
= -I(T(r)) grad T(r) (4)
whenever one admits that the conductor remains in the solid state.
The imposed data of the problem are the intensity of the current and the thermal conditions.
The resistance (l§ V2)/1, varying largely with the temperature distribution, cannot be an imposed parameter.
2.2. BASIC EQUATIONS. We have to have (nom rot H
= J + (f
= J)
div J
= o, (5)
which leads with (3) and E
= grad V to
div(-f grad V)
= o. (6)
The thermal flux leaning the volume du limited by the surface S(du) is
/p
ni ds
= div p du ~ div p du; (7)
(du) ~~~u
du~0
this flux is equal to the power dissipated in the volume du, hence div p du
= E J du
= -fE~ du, (8)
and (see Eq.(4))
div(I grad T) + -f grad V grad V
= o. (9)
We obtain so two coupled differential equations (6) and (9) completely defined by their bound- ary conditions.
3. Consideration Whether a Certain Type of Solutions May Exist
3.I. DEFINITIONS OF A PARTICULAR TYPE OF SOLUTION. We are going to investigate
whether solutions may exist [3,4] where each isothermal surface Sm(r) of the conductor is
simultaneously an equipotential surface:
T(r)
= Tm, Vr E Sm ~ V(r)
= Vm, Vr E Sm. (lo)
These solutions should satisfy Vr, a relation of the type
V(r)
=
F(T(r)) (ii)
since if T(r) is constant, V(r) must be constant too.
In order to the be able to consider this type of solutions one has also to assume for the surfaces Si and 52 that:
V(r)
= Vi, Vr E Si ~ T(r) = Ti, Vr E Si (12)
V(r) = V2, Vr E 52 ~ T(r)
= T2, Vr E S21 (13)
it means that the temperatures T[Si(r)] and T[52(r)] are uniform. We choose the subscripts
1 and 2 so that T2 > Ti
In addition, since from (II one obtains grad V
= ~()~ grad T, (14)
relation (2) leads to
(grad T n)sj~~ ~ o; (15)
so solutions of the type (11) may only exist if the lateral surface Slat of the conductor is
thermally insulated.
We are going to show that the solution of a problem (where the conditions (12), (13) and
(15) are satisfied) is certainly of the type defined by a relation (11); excepted the intensity
of the current I, this solution depends only on the temperatures Ti and T2, as well as on the
characteristics -f(T) and I(T) of the conductor.
3.2. THE ISOTHERMAL SOLUTION. In order to obtain a basis for comparison, the conduc- tor under study will be considered in the case where the temperature is maintained uniform
T(r) w To The quantities intervening in the solution of this isothermal problem Po are marked
by "2ero" subscript: (l§, Jo, lo, So, Eo, Ro), allowing the distinction from the quantities (V, J, I, S, E, R) relative to the solution of the problem P where T(r) is not uniform.
The isothermal solution corresponds to o = div Jo
" div(-f(To)Eo)
" -'to div (grad %)
" -'to A%. (16)
The auxiliary problem defined by A%
" o and by the boundary conditions:
v0(sl(~)) W v01i v0(s2(~)) ~ v02i (g~~d v0 '£l)S,at ~ ° (17)
has only one solution. By imposing then the difference [%1 %2) which corresponds to
(10)12 ~ i12
"
I (18)
the function [%i l§(r)] is completely determined.
The various equipotentiel surfaces Son(r) which correspond to [%i %n), the field lines
orthogonal to them, as well as the current density Jo(r)
= +'to grad [l§i l§(r)]
= --fo grad l§ (r) (19)
are so perfectly well known.
For the rest of the discussion the fi~nction [l§i l§(r)] plays an important role, so we give below three examples referenced by B, C and S:
for a small bar B of uniform cross section S, oriented along the direction Ox of the current, with xi < x < x2,
l§1 l§(x)
=
~~ )~~I; (20)
-fo
for a circular hollow cylinder C(ri < r < r2) of height H along axis Oz, carrying radial currents,
~~ ~~~~ 7r~-fo~~ ~i
' ~~~~
for a spherical shell S(ri < r < r2), carrying radial currents, v~~ Vo(r)
_#
) l~i ~ ~'f° ~~~~ ~~~~
In each a suitable choice has been made for the variable r referencing the equipotential surfaces
r ~ "z" for the small bars, r ~ "radius r" for the hollow cylinders, as well as for the spherical shells. Our aim is to easily illustrate the rest of the paper through the simplicity of these
examples.
Whichever is the conductor considered, its resistance (in isothermal regime) may always be defined by
%1 %2
" Ro(Io)12 = ROT. (23)
The resistance Ro includes the factor I/-fo so that the product K
= -fo Ro (24)
(which plays an essential role in the rest of the paper) depends only on the geometry of the conductor. For examples considered, the expressions of K are so:
~~ ~~
S
~~ ~~ 2/H~~ ~~
~~ 7rri~~
~~~~
Ass~Jming isothermal regime, Ro and K
= -fo Ro can be obtained through calculation, independently of the current's intensity. On the other hand, the parameter K can always be
attained ezpertmentally through measurement of the resistance between extremities Si and 52
of the conductor, at temperature To (leading to -fo) and under very small currents.
3.3. ESTABLISHMENT OF THE GENERAL RELATIONS GOVERNING J(r), T(r) AND V(r)
3.3.I. Compatibility of the Two Basic Equations. We are going to show that relations:
-f(T) grad V
= -fo grad % (26)
1(T) grad T
= -fo G(T) grad %, (27)
which satisfy (14), give the solution of equations (6) and (9) when G(T) is properly determined.
Equation (6) is thus satisfied by div(-f grad V)
= div(-fo grad l§) "'to A%
# o (28)
while (9) becomes
imposing
dG G(T) I
fi I(T) ~ -f(T) ~ ~~~~
and
~
G~(T)
~ ~ / ~
~)ji) dT' + AP' (31)
In order to let the integrals of (31) contain all the data usefi~l to the studied system, the limit temperature Tp ought to be higher or equal to the maximal temperature attained in the conductor. The constants Tp and Ap are coupled: the boundary conditions will determine in the couple (Tp, Ap) the constant which is not imposed a priort. For a material characterized
by I(T) and -f(T), the function G~ depends then only on the temperature and on the couple (Tp, Ap), and should be written in a complete manner
G~
= G~(T, (Tp, Ap)). (32)
Nevertheless, in order to simplify the notations, we shall use G~(T) whenever the role of the
couple (Tp, Ap) is not directly apparent.
The basic equations (6) and (9) may now be blended into a unique one when G~(T) is determined by (31) which leads to two possibilities for (14) with:
dF >~~~
w = +n~~~~ j~j~n~ (33~
The two signs of (33) correspond to the different combinations of increasing or decreasing V(r)
and T(r).
3.3.2. Determination ofJ(r). From (26) we have
J(r, T(r)) + Jo(r), (34)
result already given in reference [3j.
The field lines of the problem P considered are so mingled with those related to the isother- mal problem Pol similarly to each equipotential and isothermal surface Sm of the problem P
corresponds an equipotential surface Son of the problem Po, the differences (I§ Vm) and (%i %n) being however different. For the rest of the discussion we keep the notation Sm only.
3.3.3. Determination of T(r). With G(T)
= + jG(T)j, relations (14), (26) and (33) show that
the first
member being related to the problem P while the second member
reference to the
isothermal solution
problem Po). When following a field line, the various
points of which eing referenced by a
curvilinear coordinate s (with ds =jdrj), the unique
omponents of grad T and
grad % are carried on this line nd re qual to dT/ds
d% Ids,
so at, a main here the
(s~) j~T) ~ ~~ ~ ~
/~~~~~ jGjT)j~~" '~° °~~ ~ ° ~
The values sb and sa define the
intersections
of the field line
considered with the surfaces
and Sa. This result is independent of the field lineonsidered: T(sa) and§(sa) are
uniquely
linked with the
surface Sa though the literal expressions of T(s) and %
line followed. As an xample, for the onductor with revolution mmetry whose
cross section is shown in Figure 1, the field lines R and F
the frontier F) correspond to different %R(sR)
sR and SF variables. the
qualities
R P F
~R~f
S~, j~~ fib ~~
, ,
''' '''
~Fb
,,- _
~, ,,
'~ ~Ra ~pa '~
~ ~-' ',
a' ~ ~
' '
'-
_
-'
Fig I. Diametral plane cross-section of a conductor of revolution symmetry. The equipotential
and isothermal surfaces S are indicated by dashed lines. On field lines P, R and F the potential
% is given, by means of associated ~~rvilinear coordinate sP, sR and SF, respectively, by partic- ular functions %P(sP), %R(sR) and %v(sF). The set of these particular functions is such that
%P(spb)
= %R(sRb)
= %F(spb)
= %b, where spb corresponds to the intersection of line P with the surface Sb where %
= %b.
Table I. Signs relative to the expression (36) for the different cases considered. For the expressions ($0) and (Ii ), the signs are opposite of those indicated here.
absence of existence of a temperature maximum
temperature maxima
domain considered between Si and 52 between Si and SM between SM and 52
1=I12 >0 +
1=I12<0 + +
justify the notation Vo(sb) %(sa) where the field line has not to be specified. For convenience,
one may choose a field line corresponding to a simple expression of %(s); the form of the conductor shows whether an axis of symmetry, a given direction in a plan of symmetry, etc.
may be considered for this purpose.
The suitable sign which should be used in the relation (36) depends on that of the current
intensity (1
= Ii2 Positive or negative) and on the existence or absence of a maximal temper- ature TM > T2 on an equipotential surface SM. (We recall that the extremities and 2 are
identified byT2 > Ti). The various cases are summarized in Table 1.
3.3.4. Determination of V(r). From (see (26) and (27))
grad V
=
( grad T
=
+~ grad T (38)
'f 'f
and having derivated (31) with respect to the temperature (with G~ =jGj~):
2 jG(T)j fl
=
~2), (3g)
we obtain
grad V = + ~ grad T
= ~ ~~~~
grad T
= ~ grad jG(T)j (40)
'f (G( dT
Integration along a field line, in a domain where the sign remains unchanged, results in
V(Sb) V(Sa)
= ~ (lG(T(8b))1 lG(T(Sa))1). (41)
The details of calculation show that the signs + of (36) correspond to ~ of (40) and (41); the signs to be used are so the opposite to those defined in Table I.
If there is a temperature T~ (corresponding to s~) so (see (32) G(T~, (Tp, Ap))
= °, 142)
which requires the validity of (see (31) with T~ = T~ and A~
= o)
G2(T, T~) = 2j~* jdT', (43)
relation (41) becomes
V(S) V(S~) = V(S) V~ = ~llG(T(S), T~)I -zero) (44)
leading to
(v(S) v~)~ G~(T T~)
= 2 f TdT"
(45)
When the temperature presents a maximum value TM, relation (45) with T~
= TM is the basis of several works [3,4,6] with e-g- #~ of [3] equal to G~ defined by (43). In the opposite case, when s~ is outside of the interval (si, s2), T~ is only an arbitrary temperature TR which may however lead to exact results established differently by means of G~(T, (T~, A~)) of the type defined by (31).
3.3.5. Summary and Program of Investigation. For the type of problem P considered (con-
ditions (12), (13) and (15)) we have just shown that the confusion of the equipotential and isothermal surfaces effectively occurs. These surfaces are equipotentiel surfaces of the corre-
sponding isothermal problem Po (condition (26) ). By imposing for the intensity of the current lo of the problem Po the value I of the problem P, one determines completely the solution of Po and in particular the distribution Jo(r) of its current density which is identical to that of
the problem P[Jo(r)
= J(r)].
The data of the problem P (Ti, T2, 'f(T), I(T), the shape of the conductor and the intensity of the current I carried by it) enable the determination of the couple (T~, A~) and consequently
of the function G~[T, (T~, A~)]. The spatial temperature and potential variations will then be attained by means of
fi ~rad
T " + ~f° ~rad ~fi~r~ ~35~
and
grad V
= ~ grad (G(. (40)
It should be remarked that whenever one wants [1-4] to establish uniquely a relation between the temperatures and potentials (see Eq. (45) the material's single useful characteristic is the
ratio I(T) /-f(T); on the other hand, for the prevision of the temperatures (see (35) and of the
potentials, the knowledge of I(T) and of -f(T) is needed.
4. General Relations Enabling the Determination of T(r) and V(r)
4.I. FRAME OF THE SUCCEEDING DISCUSSION. General considerations show that, when
proceeding from Si towards 52, one observes either a decrease of V(r) when 1
= Ii2 is positive,
~ or an increase of it when Ii2 is negative; on the other hand, the T(r) distribution is not influenced by the sign of L In order to simplify the rest of the discussion, we shall consider the positive I case only, a simple adaptation of the V(r) expressions allowing the determination of the results for negative I values.
In the data of the problem we distinguish two groups:
the characteristics we consider to be imposed (the temperatures Ti and T2, the functions
I(T) and -f(T) linked with the material used),
the variable characteristics (intensity ofthe current I, the shape of the conductor defining
in particular the parameter K
= -foRo).
Under these conditions, and independently of the shape of the conductor, we shall show that the spatial temperature and potential variations are obtained by means of universal functions, the single parameter of which being the product KI, whereas the maximal temperature and the potential difference (Vi lfi) being uniquely fixed by this product.
4.2. STUDY OF THE FUNCTION G~IT, (Tp, Ap)]. The sign of Ihaving been fixed, it remains
only the examination of the two temperature configurations:
first configuration (I): the temperature increases monotonically from Ti to T2,
second configuration (II): the derivative (dT/ds) is zero for a value sM of
s(si < sM < s2), where the temperature attains its maximum value T(sM)
" TM
The determination of G~ IT, (T~, Ap)] and consequently of the couple (T~, A~) occurs so in two ways. For the first configuration it is sufficient to consider (see Eq. (36) and Tab. 1):
fi~~~T,~lll
A~~~i dT
- +~f°(ii
v°2) - ~f°R°I
- KI. (46)
For the second one, one should remark, by means of
TM i(T) TM i(T)
~~
T, IGII(T, (Tp, Ap))1 ~~~
T~ IGII(T, (Tp, Ap))1
" 'f0 ((V01 V0(SM)j + (V0(SM) l§2j) ~'f0 R0i = Ki, (47)
that %(sM) does not intervene In both configurations the product KI plays an essential role.
For the second configuration relation (27) shows that, for s
= sM, Gu[T(sM)j
= Gu(TM)
should tend towards zero so as grad T m order to allow grad % and J to remain finite. The
