The model proposed in this paper is based on a 3D-FEM model because it is a recognized means to simulate the electromagnetic and thermal behavior of three-dimensional objects with complex shapes [
28,
29]. The problem under study has to be analyzed by applying a multiphysics approach, since it involves coupled electro-magnetic-thermal physics. To this end, the COMSOL
® Multiphysics package [
30] has been used. Joule power losses calculated in the electromagnetic analysis are the heat source used as input data of the thermal analysis, which allows predicting the temperature evolution and distribution in the considered domain.
The 3-D mesh applied to the analyzed geometries is composed by 3-D tetrahedra. The mesh of Model A consists of 149,959 domain elements, 34,622 boundary elements, and 8850 edge elements whereas the mesh of Model B consists of 43,016 domain elements, 13,510 boundary elements, and 3426 edge elements.
To minimize the thermal influence due to the proximity of the connector, a conductor length of 1.5 m has been modeled in the simulation for both Models A and B.
4.1. Electromagnetic Analysis
Since the supply frequency is 50 Hz, the displacement current can be neglected [
31] because the quasi-static approximation applies [
32], so Maxwell’s equations become:
and
are the divergence and rotational operators, respectively;
E (V/m) ia the electric field strength;
B (T) is the magnetic flux density;
J (A/m
2) is the electric current density, and ρ
e (C/m
3) is the free electric charge density. The charge continuity equation is also considered:
The Ohm’s law establishes the relationship between the current density and the electric field as:
where σ
e (S/m) is the electrical conductivity.
From Equation (8), the resistive or Joule power losses per unit volume (W/m
3) can be calculated as,
Since the electrical conductivity σ
e is the inverse of the resistivity ρ
e, which depends on temperature [
33,
34], it can be written as:
T is the actual temperature;
ρe,0 is the electrical resistivity measured at
T0 = 293.15 K; and α
e is the temperature coefficient. Therefore, from Equations (8) and (10), Equation (9) results in:
Resistive losses Pj are the heat source applied in the heat conduction equation detailed below, this being the linkage between the electromagnetic and thermal equations.
Table 2 summarizes the magnetic and electrical parameters applied in the 3D-FEM model.
Experimental tests carried out with different substation connectors suggest that the contact resistance is approximately twice the resistance of the connector. This value has been considered in this work.
4.2. Thermal Analysis
The well-known three-dimensional heat conduction equation can be expressed as [
35]:
ρ (kg/m3) is the volumetric mass density; Cp (J/(kg·K)) is the specific heat capacity; and (W/m2) is the heat flux density. The term (W/m3) represents the specific power loss due to the Joule effect, that is, the heat source which is expressed as in Equation (11).
The link between the temperature gradient and the heat flux density is provided by the Fourier’s law of heat conduction:
k (W/(m·K)) is the thermal conductivity of the considered material. By combining Equations (11)–(13), the heat conduction equation results in [
36]:
The initial temperature condition for Equation (14) is expressed as:
where
f (x,y,z) is the initial (
t = 0) temperature distribution in the considered domain.
The natural convection and radiation boundary conditions for Equation (14), can be expressed as [
37]:
is the unit vector normal to the boundary of the analyzed domain; h (W/(m2·K)) is the convection coefficient; T∞ (K) is the air temperature; T (K) is the surface temperature; ε is the dimensionless emissivity coefficient; and σ (W/(m2·K4)) is the Stefan–Boltzmann constant. To calculate the surface-to-ambient radiation, it is assumed that the ambient behaves as a black body at the temperature T∞.
Table 3 summarizes the thermal parameters applied in the 3D-FEM model.
4.3. Heat Transfer Coefficients
This paper assumes that the cooling effect contribution is due to the thermal radiation and natural convection, although forced convection is also possible but not applied during the experimental tests. The heat transfer due to convection is often based on coefficients obtained empirically since it is a complex phenomenon and depends upon several variables such as surface dimensions and shape, flow regime, fluid temperature and properties like density, specific heat, thermal conductivity or kinematic viscosity, among others [
38,
39]. Diverse heat transfer correlations for isothermal surfaces of the most common geometries are found in [
40,
41]. Since the surfaces of the conductor and connector are not isothermal during the short-circuit evolution, this paper deals with heat transfer coefficients that change with temperature, so during simulations they are reevaluated at each time step.
The Nusselt number of Kuehn and Goldstein [
42] has been used for the horizontal cylindrical surfaces of the connectors and the conductors:
A being calculated as:
where
RaLc is the dimensionless Rayleigh number, which depends on the characteristic length
Lc (m); and
Pr is the dimensionless Prandtl number defined below. Note that for the surface of the conductors and the barrels of the connector,
Lc is the diameter of the cylinder and, for the surface of the connector,
Lc corresponds to the ratio between the surface area and the perimeter.
The Nusselt numbers of McAdams [
43] have been applied for the remaining surfaces, since they have been modelled as flat surfaces with downward and upward cooling. According to McAdams, the Nusselt number for downward cooling must be calculated as:
Note that Equation (19) has been used in the connectors’ bottom parts (Model A: the body of the connector; Model B: palm’s surfaces).
The McAdams’ Nusselt number for upward cooling is expressed as:
which has been applied to the upper parts of the connectors (Model A: caps; Model B: palms’ upper surfaces).
From the dimensionless Nusselt number, the characteristic length
Lc (m) and the thermal conductivity
k (W/(m·K)), the convective coefficient
h can be calculated as [
44]:
From the dimensionless Prandtl and Grashof numbers, one can calculate the Rayleigh number as:
whereas the dimensionless Prandtl number is obtained as:
and the Grashof number is:
Cp (J/(kg·K)) is the specific heat of air; k (W/(m·K)) is its thermal conductivity, µ (Pa·s) is the dynamic viscosity of air; g (m/s2) is the gravity of earth; β (1/K) is the thermal expansion coefficient, ρ (kg/m3) is the air volumetric mass density, Tw (K) is the surface temperature; and T∞ (K) is the fluid temperature far from the object’s surface.
Air properties such as µ, ρ and
k change with the temperature
Tfilm of the air film, so they are taken from values tabulated in [
45] and updated at each time step.
Tfilm is defined as [
46]:
Emissivity ε in Equation (16) plays a key role in calculating the radiative heat exchange. It is known that emissivity highly depends upon the condition and aging of the radiating surface, although its exact value is often difficult to determine. It is known that, for aluminum conductors, emissivity lies in the range of 0.2–0.9 [
47]. Emissivity values considered in this paper are summarized in
Table 4.
It is noted that cooling contribution due to convection and radiation during the fast short-circuit phase (0.3 s and 1 s for Model A, and 2.275 s for Model B) is very low compared with the power generated by Joule effect. However, the convective and the radiative heat flux have been calculated and taken into account for the entire duration of the test, because convective and radiative phenomena are significant during the cooling phase, that is, once the short-circuit phase has finished.