Applied Thermal Engineering 31 (2011) 3420e3427
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Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
Conjugate heat transfer in a bimetallic conductor with variable electric resistivity
O. Chávez, F. Méndez*
Facultad de Ingeniería, UNAM, Avenida Universidad 3000, México D. F. 04510, Mexico
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 7 February 2011
Accepted 17 June 2011
Available online 28 June 2011
In this work, we analyze theoretically the conjugate heat conductive mechanism resulting from an
alternating electrical current that flows continuously in an aluminum conductor steel reinforced (ACSR),
taking into account that the electric resistivity is dependent on temperature. This last consideration
conducts us to analyze simultaneously the electrical and thermal effects in both conductor materials. In
this manner, we need to solve a double conjugate thermo-electric model. In addition, the presence of
skin effect causes significant radial temperature differences, since for high frequencies the electric
current tends to flow over the surface of the conductor and therefore, the heat generation produced by
Joule effect, is no uniform. Based on an equation for predicting the alternating current density and
described by a non-linear wave equation, numerical solutions for the above equation together with the
heat conduction equation are possible to predict the current density and the temperature profiles. In
particular, the influence of the environmental convective conditions, the skin effect and the influence of
the variable resistivity on current density and temperature fields, are clarified. The above effects show
that the electrical and thermal operation of the electrical networks can be subject to several factors that
made, in general, more difficult the full utilization of the electrical transport.
Ó 2011 Elsevier Ltd. All rights reserved.
Keywords:
Ampacity
Bimetallic conductor
Conjugate model
Skin effect
Thermal behavior
1. Introduction
It is well-known that the necessity for electric power is growing
in the entire world, but economic pressures prompt the fuller
utilization of overhead transmission lines, rather than build new
lines. For this reason is really important to improve the estimations
associated to the transmission line ampacity of electrical conductors, since this parameter determines the maximum amount of
electric current admissible without damaging the conductor. One
manner to carry out the above is by predicting more accurate
temperature profiles for benefit the fuller utilization of overhead
transmission lines; therefore, we require including the simultaneous presence of different physical effects to reach a more realistic
description of the phenomenon. In the past, an analytical and
experimental pioneer study developed by Davis [1] of the thermal
behavior in cables was carried out by measuring the conductor
temperature under meteorological conditions in real-time systems.
However, temperature variations within conductor were not taken
into account. Thus, it is difficult to assess the real capacity of the
conductor. Other analysis following similar arguments and based
on a uniform temperature approximation has been considered in
* Corresponding author. Tel.: þ52 55 56228103; fax: þ52 55 56228106.
E-mail address: fmendez@servidor.unam.mx (F. Méndez).
1359-4311/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.applthermaleng.2011.06.027
the past by Black and Byrd [2]. Performing an energy balance
between the generated heat due to the Joule effect and the removed
heat by environmental conditions, these authors illustrated, in
a simple manner, a thermal model that does not consider the
existence of temperature gradients.
On the other hand, in stranded cables the gradients of temperature are larger than the corresponding gradients in solid
conductors, due to air trapped between the wires forming the
cable. Morgan [3] shows in detail the temperature profiles for the
steady-state case of a monometallic conductor, considering the
following two examples: the cable as solid and as well as a stranded
body, under the assumption of a uniform current density. He also
developed a novel analysis for calculating the equivalent thermal
conductivity for stranded cables. In parallel, Black et al. [4] developed a comparison between the temperature gradients obtained
for a steady regime with the results of a lumped model, showing
that the temperature of the center of the conductor is always larger
than the temperature calculated on the assumption of constant
temperature.
In the past, many mathematical models have been developed to
demonstrate the influence of the thermal behavior in the design of
overhead-line conductors. In this direction, early models were
made under the assumption that electrical current is uniform in the
cross section of the conductor. However, for a better representation
of the thermal behavior, it is necessary to take into account the
O. Chávez, F. Méndez / Applied Thermal Engineering 31 (2011) 3420e3427
electromagnetic effects, which affect seriously the electrical
performance. The above is particularly valid when alternating
current flows through a conductor and a redistribution of current
density is developed [5]. However, thermal effects in overhead
conductors have not been sufficiently considered in the specialized
literature and other effects related basically with mechanical
behavior of the involved materials have only been taken into
account. In this direction, some recent contributions clarify some
aspects. Liu and Findlay developed an integrated model [6] to
investigate electromagnetic, thermal and mechanical properties in
overhead conductors; however, thermal effects were only taken as
parametric by adopting very specific experimental data. Azevedo
et al. [7] showed that the performance optimization of overhead
conductors depends strongly on the fretting fatigue mechanisms. In
particular, the rupture of the planar fracture surfaces observed in
the external Aluminum strands of the conductor tested under
bending amplitude occurred by fatigue cracking followed by shear
overload. However, the range of the fatigue tests was conducted by
considering isothermal conditions for the conductor. Other relevant
mechanical aspect was analyzed by Rawlins [8], who studied the
internal damping of tensioned and overhead cables during flexure
by transverse vibration. The flexure causes relative movements
between the strands of the conductor and these movements are
constrained by friction between them. Recently, Li et al. [9] studied
statistically the reliability index and safety index of an electric
distribution system composed by a set of overhead conductors.
Taking into account various alternative technologies, these authors
showed the optimal relationship between the covered rates and the
reliability indices including public safety.
Therefore, in the present work we develop a mathematical
model for study the conjugate heat transfer process in a bimetallic
conductor in which the presence of simultaneous electromagnetic
and thermal effects has been taken into account. The case of
a variable resistivity and dependent on the temperature was
considered. Due to that the bimetallic conductor is composed by
two elements, we propose a linear relationship between electric
resistivity and temperature for both materials. The corresponding
non-linear dimensionless governing equations are solved by finite
difference method. In this manner, the temperature profiles influenced by the electromagnetic effects are obtained for realistic cases
and the influence of the environmental conditions is clarified.
Finally, we have defined a dimensionless efficiency in order to
clarify the physical influence of the most relevant parameters.
2. Description of the phenomenon
The physical model under study is shown in Fig. 1. A bimetallic
electrical conductor is composed by a typical aluminum conductor
3421
steel reinforced (ACSR). We assume that the bimetallic conductor
has a length L and the inner region occupied by the steel is
bounded by the radius a, whereas the region occupied by the
aluminum corresponds to the interval a < r b and r is the radial
coordinate whose origin is taken from the center of the circular
*
conductor. A sudden flow of alternating current, I through the
bimetallic conductor of length L is established. Thus, an increment
of temperature is originated as an opposite effect to the current
flow, causing the well-known Joule’s effect. For large values of the
frequency associated to the alternating current, a redistribution of
the current is inevitable and the skin effect yields a tendency of
the electric current to flow over the surface of the conductor. We
anticipate that this effect, among others, can seriously be affected
by the following factors: a) environmental conditions, here characterized by a uniform convective coefficient h and an ambient
temperature TN ; b) and the use of variable electrical resistivity for
both element (steel and aluminum). In general, these conductors
can be made with different materials which are constructed from
bundles of parallel conductors (strands). For simplicity, the
geometrical complexity derived from the above physical configuration is represented here by a porous medium for each
conductor with effective properties given below. In addition, we
assume that both materials satisfy the following conditions: the
aluminum works as a material with low electrical resistance,
while the steel operates as a material with high mechanical
strength. A coupled model between the governing equations for
describing the electrical phenomenon as well as the thermal
behavior is used to get a better understanding of the temperature
fields in both materials.
2.1. Electromagnetic model
From the well-known Maxwell’s equations, we can readily
derive a wave equation for analyzing the electromagnetic propagation. For simplicity, the details are omitted and can be found
elsewhere [10]. Therefore, the current density is governed by the
following equation:
*!
*
vJ
v2 l J
*
;
V lJ ¼ m
þg 2
vt
vt
2
(1)
*
In the above equation, l is the electric resistivity, J is the current
density, m is the magnetic permeability, g is the electric permittivity
and t is the physical time. From the above electrical properties, the
electric resistivity l is more sensitive to temperature variations.
Therefore, in the present work, we use a linear relationship
between this property and the temperature field, given below.
In addition, we assume that Eq. (1) is based on the fact that only
exist spatial variations of the current density in the radial direction
and the alternating current behaves like a sinusoidal wave. There*
fore, the current density can be written as J ¼ Js ðrÞeiut . On the
other hand, the linear relationship between the electrical resistivity
and the temperature field is given by as l ¼ lN ½1 þ fðT TN Þ.
Introducing it in Eq. (1), we obtain the following equation for the
steel core,
d2 Js;st
2fst
vTst 1 dJs;st
þ
þ
1 þ fst ðTst TN Þ vr
dr
r
dr 2
"
#
2
fst
v Tst 1 vTst
Js;st
þ
þ
r vr
1 þ fst ðTst TN Þ vr 2
¼
Fig. 1. Physical model of an ACSR conductor.
2i
d2st ð1
Js;st ;
þ fst ðTst TN ÞÞ
and for the aluminum,
ð2Þ
ID
647681
Title
Conjugateheattransferinabimetallicconductorwithvariableelectricresistivity
http://fulltext.study/article/647681
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