temperature and current density distribution in a bimetallic...

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

[email protected] and [email protected]

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.

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 steady-state 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

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.

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.

Recent Researches in Energy & Environment

ISBN: 978-960-474-274-5 81

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.

Fig. 1. Representation of an ACSR conductor

2.1 Electromagnetic model

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]:

22

2,

J JJ

t t

λλ µ γ

∂ ∂∇ = +

∂ ∂

��

��

(1)

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 ( ) i t

sJ J r e ω=��

. On the other hand, the electrical resistivity have a linear variation with temperature which can be written as ( )1 T Tλ λ φ∞ ∞= + − , and

introducing it into (1) we obtain the following equation for the steel core,

( )

( )

( )( )

2, ,

2

2

,2

,2

2 1

1

1

1

2,

1

s ac s acac ac

ac ac

ac ac ac

s ac

ac ac

s ac

ac ac ac

d J dJT

T T r r drdr

T TJ

T T r rr

iJ

T T

φ

φ

φ

φ

δ φ

∞

∞

∞

∂+ + + + − ∂

∂ ∂+ + =

+ − ∂∂

=+ −

(2)

In the same form we get the aluminum equation,

( )

( )

( )( )

2, ,

2

2

,2

,2

2 1

1

1

1

2,

1

s al s alal al

al al

al al al

s al

al al

s al

al al al

d J dJT

T T r r drdr

T TJ

T T r rr

iJ

T T

φ

φ

φ

φ

δ φ

∞

∞

∞

∂+ + + + − ∂

∂ ∂+ + =

+ − ∂∂

=+ −

(3)

where r is the radial coordinate, T is the temperature, T∞ is the environment temperature,

sJ 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

1/2(2 / )δ λ ωµ= , 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:

0r = : , 0s acJ

r

∂=

∂ (4)

r a= : ( )

( )

, ,

, ,

1

1

ac ac ac s ac

al al al s al

T T J

T T J

λ φ

λ φ

∞ ∞

∞ ∞

+ − =

= + − (5)

and

( )

( )

,, , ,

,, , ,

11

11

s ac ac

ac ac ac ac ac s ac

ac

s al al

al al al al al s al

al

J TT T J

r r

J TT T J

r r

λ φ λ φµ

λ φ λ φµ

∞ ∞ ∞

∞ ∞ ∞

∂ ∂+ − + = ∂ ∂

∂ ∂= + − + ∂ ∂

(6)

at the surface r b= : ,s al bJ J= . (7)

Here, a represents the radius of the steel core, while b means the total radius of the conductor, finally, bJ is the current density at the surface of the conductor and should be determinate with the following restriction:

( ), ,02

a b

s ac s ala

I J rdr J rdrπ= +∫ ∫ (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:


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( )

( )

2,

, ,

,

1

,

ac ef ac

ac ac ac s ac

ac

ac ef

k Tr T T J

r r r

Tc

t

λ φ

ρ

∞ ∞

∂∂ + + − = ∂ ∂

∂=

∂

(9)

and for aluminum:

( )

( )

2,

, ,

,

1al ef al

al al al s al

al

al ef

k Tr T T J

r r r

Tc

t

λ φ

ρ

∞ ∞

∂∂ + + − = ∂ ∂

∂=

∂

(10)

subject to the following boundary and initial conditions:

at 0 :r = 0acT

r

∂=

∂, (11)

at :r a= al acT T= , (12)

, ,al ac

al ef ac ef

T Tk k

r r

∂ ∂− = −

∂ ∂, (13)

and at the surface :r b=

( ),al

al ef al

Tk h T T

r∞

∂− = −

∂. (14)

Also it is necessary an initial condition. Here we have considered that the conductor is initially at environmental temperature.

0 :t = al acT T T∞= = . (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 analysis and

dimensionless variables 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

( ) ( ),C al eft c b a hρ −∼ . On the other hand, the suitable

spatial scale of the steel domain corresponds directly to the radius of the steel core

acr a∼ , while, the

corresponding spatial scale corresponds to the thickness of aluminum

alr b a−∼ . Furthermore, the

characteristic temperature drop cT∆ can be obtained

through an energy balance between the heat generation term and the transient term, i.e.:

( )22

,

,

al b

c

al ef

J b aT

k Bi

λ∞ −∆ ∼ , (16)

and the dimensionless parameter Bi in the above relationship represents the Biot number defined as

( )

,

.al ef

h b aBi

k

−= (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

( ) ( ),al ef

t

c b a hτ

ρ= −

, ,,

al ac

al ac

c

T T

Tθ ∞−

=∆

, r

aξ = ,

( ), ,

,

s al ac

al ac

b

J

Jϕ = ,

r a

b aη

−=

−, , ,al ac al ac c

Tκ φ= ∆ ,

ac

aca

δε = , al

alb a

δε =

−, where ( ), ,

,,

2al ac

al ac

al ac

λδ

µ ω

∞= .

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: a bχ = .

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:

( )

( ) ( )

2

2

2

2 2

12

1

1 2,

1 1

ac ac ac ac

ac ac

ac ac ac

ac ac

ac ac ac ac ac

d d

dd

i

ϕ κ θ ϕ

κ θ ξ ξ ξξ

κ θ θϕ ϕ

κ θ ξ ξξ κ θ ε

∂+ + + + ∂

∂ ∂+ + =

+ ∂∂ +

(18)

and aluminum dimensionless model:

( )

( )

( )

2

2

2

2

2

12

11

1

11

2,

1

al al al al

al al

al al al

al

al al

al

al al al

d d

dd

i

ϕ κ θ ϕ

χκ θ η ηη ηχ

κ θ θϕ

χκ θ ηη ηχ

ϕκ θ ε

∂ + + +

+ ∂ + −

∂ ∂ + + =

+ ∂∂ + −

=+

(19)

with their dimensionless boundary conditions:

0ξ = : 0acdϕ

ξ=

∂, (20)

1 0ξ η− = = : [ ][ ]

,

,

1

1al al al

ac al

ac ac ac

λ κ θϕ ϕ

λ κ θ∞

∞

+=

+, (21)


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( )( )

,

,

1

1 1

,1 1

ac ac acal ac ac ac

ac

ac al al ac ac ac

al al al

al

al al

d

d

d

λ κ θµ ϕ κ θϕ

µ λ κ θ ξ κ θ ξ

ϕ κ θ χϕ

η κ θ η χ

∞

∞

+ ∂+ =

+ + ∂

∂ = +

∂ + ∂ −

(22)

1η = : 1

alϕ = . (23)

2.5 Dimensionless thermal model 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:

( )

( )

( )

22,,

, ,

2

, ,

, ,

11

1

,1

al efacac

ac ac ac

al ac ef

al ef ac ef ac

ac ef al ef

kBi

k

ckBi

k c

λθ χξ κ θ ϕ

ξ ξ ξ λ χ

ρ θχ

ρ χ τ

∞

∞

∂ ∂+ + =

∂ ∂ −

∂ =

− ∂

(24)

aluminum dimensionless model:

( )21

11

1

.

al

al al al

al

Bi

Bi

θχη κ θ ϕ

χ η χ ηηχ

θ

τ

∂ ∂+ + + =

∂ − ∂ +−

∂=

∂

(25)

together with the associated boundary and initial conditions given above.

0 :ξ = 0acθ

ξ

∂=

∂, (26)

1 0 :ξ η− = =

,

,

1ac ef ac al

al ef

k

k

θ θχ

χ ξ η

∂ ∂ −=

∂ ∂ , (27)

ac alθ θ= (28)

1:η = al

alBi

θθ

η

∂= −

∂, (29)

0 :τ = 0ac al

θ θ= = . (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 ,

alε ,

acε ,

alκ and

acκ ; nevertheless we can reduce the number of

parameters considering a constant Biot number 1Bi = , also, the skin parameter for aluminum

alε is related

with the skin parameter of steel ac

ε , because the

angular frequency is the same in both cases, In the same manner the coupling parameter for the aluminum media

alκ is related with the coupling

parameter for the steel core, because both parameters have the same characteristic temperature drop

cT∆ , so

they are related with the following expressions:

,

,

1 ac al

ac al

al ac

λ µχε ε

χ λ µ∞

∞

−=

and ac

ac al

al

φκ κ

φ= .

For the above reasons now we have two main parameters that affect the mathematical model:

alε

and al

κ . From now on al

ε and al

κ 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 (steel-aluminum), 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. 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 parameterε (=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.

Fig. 2. Variation of dimensionless current density distribution, ϕ ,

for different values of ε

For the following Figs. 3-5, we show the results for

the heat transfer process following this sequence: in


ISBN: 978-960-474-274-5 84

order to see the influence of skin effect on the dimensionless temperature profiles, we choose a fixed values of the Biot number 1Bi = , and 0.5κ = , 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. 3. Dimensionless temperature profile,θ , for different values of the dimensionless parameter ε and τ =0.1.



3.2 Influence of κ In Fig. 6, we show the influence of κ on the current density distribution keeping invariable the parameters

1.0Bi = and 0.5ε = for three different values of the parameter κ (=0.0, 0.2, 0.5). It is worth to mention that for the case 0.0κ = is the solution of the uncoupled model, in this case is possible to solve the electromagnetic and thermal models separately.

Fig. 6. Variation of dimensionless current density distribution, ϕ ,

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

1.0Bi = and 0.5ε = 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.


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Fig. 7. Dimensionless temperature profile,θ , for different values of the dimensionless parameter κ and τ =0.1.



4 Conclusions In summary, the temperature profiles are strongly

influenced by the following parameters: Bi , κ ,andε . 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,

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 0κ ≠ , 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.

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.


ISBN: 978-960-474-274-5 86

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