Recent Researches in Energy & Environment
Temperature and Current Density Distribution in a Bimetallic
Conductor Through a Coupled Model
OSCAR CHÁVEZ and FEDERICO MÉNDEZ
Facultad de Ingeniería
Universidad Nacional Autónoma de México
Insurgentes Sur 3000, D. F.
MÉXICO
aochvzlpz@yahoo.com.mx and fmendez@servidor.unam.mx
Abstract: - In this paper, we analyze theoretically the conduction heat transfer that results from an alternating
electrical current that flows continuously in an aluminum conductor steel reinforced (ACSR), taking into account
that the electrical resistivity is a linear function of temperature. This effect, necessarily produce a situation that
forces us to analyze simultaneously the electrical and thermal effects in both materials. In this way we obtain a
thermo-electric model. In this way, and by simultaneous numerical solution of Maxwell's equations and heat
diffusion equation, it is possible to determine both the current density distribution and the temperature gradients
during its transient state. Also, we show the influence of the steel core on the current density and temperature
distributions. The results are presented in dimensionless form, in order to reduce the number of physical variables
involved. In particular, we show the influence of the skin effect and the variation of the resistivity with temperature
on current density and temperature gradients. The above shows that the electrical operation of the networks is
subject to several factors that may make more difficult the full utilization of electrical transport.
Key-Words: - Ampacity, Bimetallic conductor, Coupled model, Skin effect, Thermal behavior.
comparison betwe−en the temperature gradients in a
steady regime with the results obtained with the aid of
a lumped model, showing that the temperature of the
center of the conductors is always higher than the
temperature calculated on the assumption of constant
temperature.
Early models were made under the assumption that
electrical current is uniform in the cross section of the
conductor. However, In order to have a better
representation of the thermal behavior is necessary to
take into account the 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].
Therefore, in the present work we develop a
mathematical model for a bimetallic conductor in
which the presence of simultaneous electromagnetic
and thermal effects has been taken into account.
1 Introduction
Early analytical and experimental studies of the
thermal behavior in cables were carried out measuring
conductor temperature and meteorological conditions
in real-time systems, which did not take into account
temperature variations within conductor, for this
reason it does not allow using the full function of the
conductor [1].
Normally, a uniform temperature approximation
had been considered, and it was determined by an
energy balance between the generated heat due to the
Joule effect and the removed heat by environmental
conditions. Reference [2] shows a thermal model that
does not consider the existence of temperature
gradients and was solved by the transient and steadystate cases.
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. Reference [3]
shows in detail the temperature profile for the steady –
state case of monometallic cables, considering both
cases: the cable as solid and as well as a stranded
solid, under the assumption of uniform current
density. He also developed a novel analysis to
calculate the equivalent thermal conductivity for
stranded cables. Reference [4] developed a
ISBN: 978-960-474-274-5
2 Description of the phenomenon
The case study is a sudden flow of alternating
current through a bimetallic conductor, in this case an
ACSR conductor showed in fig. 1, where the shaded
−
This work was supported by Consejo Nacional de Ciencia
y Tecnología at Mexico, under the contract number 79811.
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Recent Researches in Energy & Environment
area represents the steel core; on the other hand, the
unshaded area represents the aluminum conductor. An
increase of temperature is originated for the
opposition to the current flow, this phenomenon is
called the Joule’s effect, since alternating current
flows through the ACSR conductor, redistribution is
presented due the skin effect, which consists on the
tendency of the electric current to flow over the
surface of the conductor.
In the same form we get the aluminum equation,

2φal
∂Tal 1  dJ s , al
+
+ 
+


dr
 1 + φal (Tal − T∞ ) ∂r r  dr
 ∂ 2Tal 1 ∂Tal 
φal
+
+

 J s , al =
1 + φal (Tal − T∞ )  ∂r 2 r ∂r 
d 2 J s , al
2
=
2i
J s , al ,
δ (1 + φal (Tal − T∞ ) )
2
al
where r is the radial coordinate, T is the temperature,
T∞ is the environment temperature, J s is the current
density function depending only on radial coordinate,
φ is the temperature coefficient for resistivity, λ∞ is
the resistivity at environment temperature and i is the
imaginary number − 1 .
In addition, we introduced the well-known
conductor skin depth parameter, δ , defined by
δ = (2λ / ωµ )1/ 2 , where ω is the frequency of the
electrical signal, which can be found elsewhere [7].
The above equations system must be solved with
the following boundary conditions:
Fig. 1. Representation of an ACSR conductor
∂J s , ac
r =0:
2.1
Electromagnetic model
∂t
dr

φac
I = 2π
= λ∞ , al 1 + φal (Tal − T∞ )  J s , al
(5)
(∫ J
a
0
b
s , ac
rdr + ∫ J s , al rdr
a
)
(8)
where I is the total circulating electrical current. In
our case, we assume known values for this variable
given lines below.
2.2 Thermal model
In order to determinate the gradients of
temperature generated by the Joule effect, we must
solve the heat conduction equation in transient state
for each one of the media of the conductor, regarding
only temperature variations in the radial coordinate.
For steel domain:
2
ISBN: 978-960-474-274-5
λ∞ , ac 1 + φac ( Tac − T∞ )  J s , ac =
Here, a represents the radius of the steel core,
while b means the total radius of the conductor,
finally, J b is the current density at the surface of the
conductor and should be determinate with the
following restriction:

2φac
∂Tac 1  dJ s , ac
+
+
+
 1 + φ (T − T ) ∂r r  dr
ac
ac
∞


 ∂ Tac 1 ∂Tac 
+

 J s , ac =
1 + φac (Tac − T∞ )  ∂r 2
r ∂r 
2i
= 2
J s , ac ,
δ ac (1 + φac (Tac − T∞ ) )
+
(4)
∂J s , ac
∂T 
1 
+ λ∞ , acφac J s , ac ac  =
 λ∞ , ac 1 + φac ( Tac − T∞ ) 
µ ac 
∂r
∂r 
(6)
∂J s , al
∂Tal 
1 
=
+ λ∞ , al φal J s , al
 λ 1 + φal (Tal − T∞ ) 

µal  ∞ , al 
∂r
∂r 
J s , al = J b .
at the surface r = b :
(7)
introducing it into (1) we obtain the following
equation for the steel core,
2
=0
and
In the above equation, λ is the electric resistivity,
J is the current density, µ is the magnetic
permeability, γ is the electric permittivity and t is the
physical time.
We consider that exist only variations of the
current density in the radial direction and the
alternating current behaves like a sinusoidal wave.
Therefore, the current density can write
as J = J s (r )eiωt . On the other hand, the electrical
resistivity have a linear variation with temperature
which can be written as λ = λ∞ 1 + φ (T − T∞ ) , and
d 2 J s , ac
∂r
r =a:
From the well-known Maxwell’s equations, we can
readily derive a wave equation to analyze the
electromagnetic propagation. Therefore, the current
density is governed by the following equation [6]:
 ∂J
∂ 2λ J 
,
∇2λ J = µ 
+γ
(1)
2 
 ∂t
(3)
(2)
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Recent Researches in Energy & Environment
kac ,ef ∂  ∂Tac 
+ λ∞ , ac 1 + φac (Tac − T∞ )  J s , ac
r
r ∂r  ∂r 
∂T
= ( ρ c )ac ,ef ac ,
∂t
2
=
Bi =

 + λ∞ , al 1 + φal (Tal − T∞ )  J s , al

2
r = 0:
=
at
r =a:
∂Tac
= 0,
∂r
Tal = Tac ,
− kal , ef
∂Tal
∂T
= −kac ,ef ac ,
∂r
∂r
∂Tal
= h (Tal − T∞ ) .
∂r
Tal , ac − T∞
δ ac
δ al
ξ=
r
,
a
κ al , ac = φal , ac ∆Tc ,
2λ∞ ,( al , ac )
ε ac =
(12)
It is worth to point out that the subscript al , ac
means that a variable o parameter can be used
whereas aluminum or steel. Also it is necessary to
define a geometrical parameter: χ = a b .
(13)
a
ε al =
,
(11)
,
b−a
, where δ al , ac =
µ al , acω
.
(14)
analysis
2.4 Dimensionless electromagnetic model
The system (2)-(7) can be written in dimensionless
form by using the above dimensionless parameters
and variables.
Steel core dimensionless model:
∂θ ac 1  dϕac
d 2ϕ ac 
κ ac
+  2
+ 
+
2
dξ
 (1 + κ acθ ac ) ∂ξ ξ  d ξ
 ∂ 2θ ac 1 ∂θ ac
κ ac
+
+

(1 + κ acθ ac )  ∂ξ 2 ξ ∂ξ

2i
ϕ ac ,
 ϕac =
2
1
+
κ
(

acθ ac ) ε ac
(18)
and aluminum dimensionless model:

d 2ϕ al 
κ al
∂θ al
1
+2
+
2
dη
 (1 + κ alθ al ) ∂η η + χ

1− χ

and
In order to reduce the number of physical
parameters and variables, we can develop an order of
magnitude analysis. First, we identify various scales:
the
characteristic
convective
time
scale
tC ∼ ( ρ c ) al , ef ( b − a ) h . On the other hand, the suitable


 dϕ
 al +
 dη



 ∂ 2θ
κ al
∂θ al 
1
al

ϕ =
+
+
2
(1 + κ alθ al )  ∂η η + χ ∂η  al


1− χ


spatial scale of the steel domain corresponds directly
to the radius of the steel core rac ∼ a , while, the
corresponding spatial scale corresponds to the
thickness of aluminum ral ∼ b − a . Furthermore, the
characteristic temperature drop ∆Tc can be obtained
through an energy balance between the heat
generation term and the transient term, i.e.:
2
λ∞ , al J b2 ( b − a )
,
(16)
∆Tc ∼
=
2i
(1 + κ alθ al ) ε al2
and the dimensionless parameter Bi in the above
relationship represents the Biot number defined as
83
(19)
ϕal ,
with their dimensionless boundary conditions:
dϕ ac
= 0,
ξ =0:
∂ξ
λ [1 + κ alθ al ]
ϕ ac = ∞ , al
ϕ ,
ξ −1 = η = 0 :
λ∞ , ac [1 + κ acθ ac ] al
kal , ef Bi
ISBN: 978-960-474-274-5
, θ al , ac =
∆Tc
( ρ c ) ( b − a ) h 
al , ef


J s ,( al , ac )
r −a
ϕ al , ac =
,
η=
,
Jb
b−a
Also it is necessary an initial condition. Here we have
considered that the conductor is initially at
environmental temperature.
Tal = Tac = T∞ .
t = 0:
(15)
In the above equations, k is the thermal
conductivity of the metallic conductor, ρ is the
density, c is specific heat, h is the convective heat
transfer coefficient and subscript ef denotes effective
properties, while subscripts al and ac , refer to the
aluminum and steel domain respectively.
2.3 Order of magnitude
dimensionless variables
t
τ=
(10)
and at the surface r = b :
− kal , ef
(17)
.
With the above set of characteristic geometrical and
physical scales, the electromagnetic and thermal
models can be considerably simplified by introducing
the following dimensionless variables
subject to the following boundary and initial
conditions:
at
kal ,ef
(9)
and for aluminum:
kal , ef ∂  ∂Tal
r
r ∂r  ∂r
∂T
= ( ρ c ) al , ef al
∂t
h (b − a )
(20)
(21)
Recent Researches in Energy & Environment

µ al λ∞ , ac (1 + κ acθ ac )  dϕ ac
κ ac ∂θ ac
+
ϕ ac  =

µ ac λ∞ , al (1 + κ alθ ac )  d ξ 1 + κ acθ ac ∂ξ

 dϕ
 χ 
κ al ∂θ al
=  al +
ϕ al  
,
 ∂η 1 + κ alθ al ∂η
1− χ 
η =1:
ϕ al = 1 .
angular frequency is the same in both cases, In the
same manner the coupling parameter for the
aluminum media κ al is related with the coupling
parameter for the steel core, because both parameters
have the same characteristic temperature drop ∆Tc , so
they are related with the following expressions:
(22)
(23)
φ
 1 − χ  λ∞ , ac µal
and κ ac = κ al ac .

φal
 χ  λ∞ , al µ ac
ε ac = ε al 
2.5 Dimensionless thermal model
For the above reasons now we have two main
parameters that affect the mathematical model: ε al
and κ al . From now on ε al and κ al will be called
simply ε and κ . which measures the environmental
conditions, electrical thickness of the current density
and the level of coupling between both models –
electric and thermal models- respectively.
In particular, we show the current densities and the
temperature profiles as functions of the radial
coordinate of each one of the domains (steelaluminum), on the left part of each figure is presented
the solution of the equation corresponding to the steel,
on the other hand, the right part of each figure shows
the solution of the aluminum equation.
In the same manner, we can use the dimensionless
variables and parameters in order to obtain the
following dimensionless thermal model:
Steel core dimensionless model:
2
1 ∂  ∂θac  λ∞,ac kal ,ef  χ 
2
+
ξ



 (1 + κ acθac ) Bi ϕac =
ξ ∂ξ  ∂ξ  λ∞,al kac,ef  1 − χ 
(24)
2
kal ,ef ( ρ c )ac ,ef  χ 
∂θ ac
=
,

 Bi
kac ,ef ( ρ c )al ,ef  1 − χ 
∂τ
aluminum dimensionless model:
∂ 
1
χ  ∂θ al 
2
η+
+ Bi (1 + κ alθ al ) ϕ al
χ ∂η  1 − χ  ∂η 
η+
1− χ
∂θ
= Bi al .
∂τ
together with the associated boundary and
conditions given above.
∂θ ac
=0,
ξ = 0:
∂ξ
ξ −1 = η = 0 :
kac ,ef  1 − χ  ∂θ ac ∂θ al
,
=


∂η
kal ,ef  χ  ∂ξ
θ ac = θ al
∂θ al
= − Biθ al ,
η = 1:
∂η
τ = 0:
θ ac = θ al = 0 .
=
(25)
initial
3.1 Influence of
ε
In Fig. 2 we show the current density as a function
of the dimensionless radial coordinate for three
different values of the skin parameter ε (=0.1, 0.2 and
0.5). Clearly, for small values of this parameter, the
stronger skin effect is enhanced because the current
density distribution tends to flow in regions closer to
the surface, and through the steel core practically does
not flow electric current in all cases.
(26)
(27)
(28)
(29)
(30)
3 Results
The above dimensionless heat conduction and
electromagnetic equations together with their
boundary and initial conditions, here represented by
the system (18)-(30) were solved using the
conventional finite-differences method [8].
As we can see, the set of ecs. (18)-(30) is highly
influenced by five parameter: Bi , ε al , ε ac , κ al and
κ ac ; nevertheless we can reduce the number of
parameters considering a constant Biot number Bi = 1 ,
also, the skin parameter for aluminum ε al is related
with the skin parameter of steel ε ac , because the
ISBN: 978-960-474-274-5
Fig. 2. Variation of dimensionless current density distribution,
for different values of
ϕ,
ε
For the following Figs. 3-5, we show the results for
the heat transfer process following this sequence: in
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Recent Researches in Energy & Environment
order to see the influence of skin effect on the
dimensionless temperature profiles, we choose a fixed
values of the Biot number Bi = 1 , and κ = 0.5 ,
different values of the dimensionless time τ (=0.1,
1.0 and 10.0) and three different values of the skin
parameter ε (=0.1, 0.2, 0.5). Therefore, Figs. 3-5
show the thermal behavior of dimensionless
temperature as a function of the dimensionless radial
coordinate. These figures reveal clearly the influence
of the skin effect: for increasing values of ε the
domain of the spatial variation of the temperature is
going to be more relevant. Together with the above
considerations, on the other hand, these figures reveal
that for increasing values of the dimensionless time
the temperature profiles tend to reach a steady-state
condition.
Fig. 5. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter ε and τ =10.0.
3.2 Influence of
κ
In Fig. 6, we show the influence of κ on the current
density distribution keeping invariable the parameters
Bi = 1.0 and ε = 0.5 for three different values of the
parameter κ (=0.0, 0.2, 0.5). It is worth to mention
that for the case κ = 0.0 is the solution of the
uncoupled model, in this case is possible to solve the
electromagnetic and thermal models separately.
Fig. 3. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter ε and τ =0.1.
Fig. 6. Variation of dimensionless current density distribution,
for different values of κ
For the following Figs. 6-9, we show the transient
state of temperature profile. In order to show the
effect of the parameter κ we kept uniform values of
the parameters
Bi = 1.0 and ε = 0.5 and varying the dimensionless
time τ (=0.1, 1.0 and 10.0). Thus, for increasing
values of κ the spatial variation of the temperature is
going to be more relevant; otherwise, we can
considerate both models separately.
Fig. 4. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter ε and τ =1.0.
ISBN: 978-960-474-274-5
ϕ,
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leading it also to an overheated of the cable, and
finally a small value of κ means that electric and
thermal model are weakly coupled and can be solved
separately. On the other hand, we observe that when
we take into account the variation of resistivity with
temperature κ ≠ 0 , the thermal behavior presents an
increase on temperature profiles; in the same manner
this increase on temperature originates redistribution
on the current density profile. The steel core
practically does not carry electric current;
nevertheless, it has a strong influence on the thermal
behavior during the transient state, because part of the
heat generated is transferred to the core by diffusion
causing a delay to reach the steady state. The above
results are very interesting from a practical point of
view, because the global electrical performance and
design of high-tension cables is seriously altered by
the influence of these factors.
Fig. 7. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter κ and τ =0.1.
References:
[1] M. W. Davis, A new thermal rating approach: the
real time thermal rating system for strategic
overhead conductor transmission lines. Part I.
General description and justification of the real
thermal rating system, IEEE Trans. on Power
Apparatus and Systems, vol. 96, No. 3, 1977, pp.
803-809.
[2] W. Z. Black and W, R, Byrd, Real-time ampacity
model for overhead lines, IEEE Trans. on Power
Apparatus and Systems, vol. 102, No. 7, 1983, pp.
2289-2293.
[3] V. T. Morgan, The radial temperature distribution
and efective radial thermal conductivity in bare
solid and stranded conductors, IEEE Trans. on
Power Delivery, vol. 5, 1990, pp. 1443-1452.
[4] W. Z. Black, S. S. Collins, and J. F. Hall,
Theoretical model for temperature gradients
within bare overhead conductors, IEEE Trans. on
Power Delivery, vol. 3, No. 2, 1988, pp. 707-715.
[5] A. Jordan, A. Barka, and M. Benmouna, Transient
state temperature distribution in a cylindrical
conductor with skin effect, Int. Journal of Heat
and Mass Transfer, vol. 30, No. 11, 1987, pp.
2446-2447.
[6] J. D. Jackson, Classical Electrodynamics, Ed.
John Wiley & Sons, Inc., USA, 1975.
[7] A. A. Ghandakly, and R. L. Curran, A model to
predict current distributions in bundled cables for
electric glass melters, IEEE Trans. on Industry
Applications, vol. 26, No. 6, 1990, pp. 1043-1048.
[8] M. Özisik, Finite Difference Methods in Heat
Transfer, CRC, USA, 1994.
Fig. 8. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter κ and τ =1.0.
Fig. 9. Dimensionless temperature profile, θ , for different values
of the dimensionless parameter κ and τ =10.0.
4 Conclusions
In summary, the temperature profiles are strongly
influenced by the following parameters: Bi , κ ,and ε .
For small values of these parameters, the physical
consequences are direct: small values of skin
parameter ε has a undesirable effect, because a great
amount of electric current flows through a small area,
ISBN: 978-960-474-274-5
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