Change in temperature distribution by constriction of electric current through contact area

Abstract

The change in temperature distribution due to constriction of an electric current through a contact area is analytically and numerically investigated. The potential distribution, current density distribution and heat density distribution inside the body are analytically obtained. The heat density has the singular point at the periphery of the contact area. It is revealed that the temperature has a finite value at the periphery, whereas the temperature change rate with respect to time has an infinite value because of the singularity. The results show the change in the temperature distribution. The times to melting and melting area are determined from the change in the temperature distribution. The total heat generated until melting occurs is also obtained.

Keywords

Change in temperature distribution Constriction of electric current Contact Analytical derivation Numerical calculation Time to melting Melting area

Introduction

Heat generation by constriction of an electric current through a contact area as shown in Fig. 1 is important for technological applications such as resistance spot welding, electric relays, and electric and electronic switches. The heat changes the conditions in the contact area, and knowledge of the conditions is important for device design.

It is apparently difficult to observe the contact area directly. The conditions in the contact area are checked using relevant methods such as measuring the electric resistance or electric current.^1–5 To understand the conditions in the contact area, it is important to understand the relation between changes in the conditions and the changes in the electric resistance or electric current. Real surfaces have surface roughness, and the actual contact areas are tiny.^6–8 The electric current flows through these tiny spots, and the heat generated by constriction of the current softens or melts the contact area. The contact area deforms, and its conditions change. Thus, heat generation around the tiny contact area is of primary importance. We focus on a single tiny spot for simplicity.

Table 1

Nomenclature

Symbols	Definitions/units
F	Arbitrary vector in the spheroidal coordinate system
	Components of the spheroidal coordinate system/.
a	Contact radius/m
i ( r )	Current density distribution/A m^− 2
	Mass density/kg m^− 3
h _j	Difference between determined by numerical calculation and asymptotic function/.
	Differential time/s
	Differential values of
	Differential volume/m³
I	Electric current passing the system per unit time/A
E ( r )	Electric field distribution/V m^− 1
	Electric resistivity/Ω m
Err_j	Error between the numerical calculation and asymptotic function/.
q( r )	Heat density distribution/J m^− 3 s^− 1
	Heat generation in unit time/J
j	Index of notation for several points of
	Initial temperature/K
	Melting temperature/K
	Time to melting/s
	Normalised /.
	Normalised /.
	Normalised /.
	Normalised /.
	Normalised T/.
	Normalised temperature distribution under the steady state condition/r'.
	Point in R
	Potential at contact area/V
	Potential distribution/V
	Radial, tangential and axial components of the cylindrical coordinate system/m,.,m
R	Space where heat is generated
C	Specific heat/J K^− 1 kg^− 1
d S	Surface element vector
	Temperature distribution/K
κ	Thermal conductivity/W m^− 1 K^− 1
D	Thermal diffusivity/m² s^− 1
t	Time/s
Q	Total heat generated until melting occurs/J
	Unit vectors in directions of the spheroidal coordinate system
	Variable of integration for time/s
	Variables introduced by the variable transformation
r	Vector of the cylindrical coordinate system

The current density distribution inside the body was analytically obtained by Greenwood and Williamson under the steady state condition, in which the temperature distribution does not change.⁹ The theory is a development of the research of Kohlrausch, who obtained the relation between the temperature and potential inside the body under the steady state condition.¹⁰ The heat density distribution is derived from the current density distribution, and the heat density has a singular point at the periphery of the contact area. In contrast, the temperature distribution under the steady state condition, which was also obtained by Greenwood and Williamson, does not have a singular point and has the same value at the contact area.

Experiments show that melting occurs around the periphery of the contact area.^{9, 11} From the analysis by Greenwood and Williamson, melting occurs before the temperature distribution reaches the steady state. Kohler and Zielasek analytically derived the temperature change, but they focused only on the temperature change at the centre of the contact area and did not discuss the change in the temperature distribution in other regions, including the periphery of the contact area.¹²

The change in the temperature distribution, including that at the periphery of the contact area, has been numerically analysed using the finite element method. However, the calculation around the singular point of the heat density distribution is not clear. Kubono removes the effect of heat generation at the point,¹³ and Robertson does not clearly mention how to treat that point.¹⁴

This research analytically and numerically investigates the change in the temperature distribution due to constriction of an electric current through the contact area. We first introduce a simple system for the analysis. Then, the change in the temperature distribution is derived. The singularity is strictly discussed. The results suggest the time to melting, melting area and total heat generated until melting occurs, which corresponds to the total energy consumption until melting occurs. Those results facilitate understanding of the change in the condition of the contact area and theoretical research on applications such as resistance spot welding.^15–17 1

Schematic illustration of constriction of electric current through contact area

Theory

System

Figure 2 shows the axially symmetric semi-infinite conductive system introduced in this research. The shaded area represents the contact area, and a is the contact radius (m). Regarding the coordinates, the origin O is set at the centre of the contact area, the axis of symmetry is the z axis, the radial direction is the r axis, and the direction tangential to the circle is the θ axis. An arbitrary point inside the body is expressed as r = (r, θ, z) . Regarding the boundary conditions, at the contact area , the potential is V_c (V); at an infinite distance from the contact area, the potential equals zero; at the upper surface of the body excluding the contact area (r>a, z = 0), the component normal to the surface of the gradient of the potential equals zero.

Schematic illustration of system

Derivation of current density distribution

If the magnetic effect is ignored, the potential distribution V( r ) (V) and electric field distribution E ( r ) (V m^− 1) satisfy the relation 1 Since there is no charge inside the body, E ( r ) satisfies Gauss's law. 2 From equations (1) and (2), the potential distribution satisfies 3 where Δ is the Laplace operator. Equation (3) is the Laplace equation. From the uniqueness of the solution of the Laplace equation, V( r ) is uniquely determined to satisfy equation (3) and the boundary conditions of the system. When V( r ) is uniquely determined, E ( r ) is uniquely determined by equation (1). From Ohm's law, the current density distribution i ( r ) (A m^− 2) and E ( r ) satisfy the following relation. 4 where ρ is the electric resistivity (Ω m). Since E ( r ) is uniquely determined, i ( r ) is uniquely determined by equation (4).

Consider V( r ) as given by 5 where ξ is a component of the spheroidal coordinate system. The parameters r, z are expressed in terms of the components of the spheroidal coordinate system ξ, η as 6a 6b respectively, where and . θ is the same for both coordinate systems. Equations (6a) and (6b) are transformed into equation (7) by eliminating η, where it is easier to understand that ξ, θ, η express the spheroid. 7 The gradient and divergent in the spheroidal coordinate system are given by 8 9 respectively, where are unit vectors in the directions. ξ, θ, η respectively and are the components of arbitrary vector F in the spheroidal coordinate system. From equations (8) and (9), the Laplace operator in the spheroidal coordinate system is derived as 10 From equation (10), equation (5) satisfies equation (3). Additionally, equation (5) satisfies all the boundary conditions mentioned in the description of the system. Therefore, equation (5) is the unique solution of the potential distribution in the system. From equations (1), 4, 5 and (8), the current density distribution is obtained as 11 Equation (5) is similar to the potential distribution obtained by Greenwood and Williamson.⁹ If V_c is subtracted from equation (5), the equations are the same. The reason for the difference is the differences in the boundary conditions. Greenwood and Williamson consider a conductor created from two large pieces of metal that touch each other over a small area, where the potential of the small area equals zero, and the potentials at an infinite distance from the contact are V_c on a surface and − V_c on the other side of the surface. If the system is separated into two parts at the small area, one piece has the same geometry as this system. As for the boundary conditions, the potential equals zero at the contact area and − V_c at an infinite distance from the contact area. The sign of the potential results from the definition of the direction of ξ. Thus, the difference in the potentials of the systems is a constant value of the potential V_c. Therefore, if V_c is subtracted from equation (5), the equations are the same. Equation (11) is exactly the same as the equation obtained by Greenwood and Williamson because the current density distribution does not depend on the constant potential difference.

If we normalise as , then 12

Derivation of temperature distribution change

The heat density distribution q( r ) (J m^− 3 s^− 1) is expressed as follows using Joule's law and equation (11). 14 If we normalise q( r ) as , then 15 If the loss of heat from the surface of the body is ignored, the thermal conduction equation is expressed as 16 where T( r , t) is the temperature distribution (K), is the thermal diffusivity (m² s^− 1), C is the specific heat (J K^− 1 kg^− 1), ρ_d is the mass density (kg m^− 3), κ is the thermal conductivity (W m^− 1 K^− 1), and t is the time (s). We define t = 0 when V_c is applied to the contact area.

Assuming that the material constants (specific heat, mass density and thermal conductivity) are uniform and constant, and that the effects of the phase transition are ignored, the change in the temperature distribution for an arbitrary heat density distribution is obtained as follows by solving equation (16). 17 Suppose T₀ is the initial temperature (K), R is the space where heat is generated, r′ is a point in R , dν′ is the differential volume, t′ is the variable of integration for time (s) ( ), and is the differential of t′. From equations (14) and (17) and , the change in the temperature distribution for the constriction of the current is obtained as follows: 18 Kohler and Zielasek obtained an equation similar to equation (18).¹² However, they derived only the equation for the temperature change at the centre of the contact region. We normalise the parameters as (if . Thus, , and . The normalised temperature distribution change is then obtained as 19 The integral range is . The range of η is and is not because the upper surface of the body is assumed to be isothermal, and heat propagates as if it is reflected at the boundary.

The temperature distribution is axially symmetric. Thus, , and .

Results and discussion

Results

Figure 3a–3c shows , and respectively obtained using equations (12), 13 and (15). The gradient of the potential distribution with respect to the radius is discontinuous at the periphery of the contact area ( ). Thus, the current density distribution and heat density distribution have a singularity at the same point. The singularity causes an infinite temperature change rate with respect to time.

a normalised potential; b normalised current density; c normalised heat density

We differentiate equation (19) with respect to and obtain 20 Note that is normalised as . has a finite value when . When , the diffusion, which is the first term of equation (16), equals zero. Thus, 21 Obviously, has a singular point at . This singularity is derived from the singularity of the current density distribution.

The temperature change rate has an infinite value at the periphery of the contact area when . However, this does not mean that the temperature has an infinite value. Consider the heat diffusion from a point where heat is input. The heat diffuses just after it is input and is widely distributed in space. Thus, the heat diffuses into a wider region as the elapsed time since the heat input increases. Therefore, if the elapsed time is short, the heat is concentrated near the point where it is input. Next, consider the heat input distribution. The temperature change rate is determined by integrating the heat diffusion from all the points where heat is input. When the elapsed time since the heat input is long, it is necessary to integrate the heat diffusion from distant points. When the elapsed time is short, the temperature change rate is estimated by integrating only the heat diffusion from the vicinity.

The singular points are . It is sufficient to calculate the temperature change rate with respect to at because of the axial symmetry. Thus, we can approximately calculate equation (20) by integrating the heat diffusion around when . We set , where ; then, is expanded in a series as 22 where O is the Landau symbol. The point corresponds to . When , the integral range can be constricted as , where satisfy . If , then is approximated as 23 Therefore, 24 where erf is the error function. If , the terms of the error function equal one. Thus, 25 for sufficiently small . From equation (25), has an infinite value at ; however, the integration of with respect to is 26 Thus, the temperature has a finite value. We think that this apparent paradoxical relation between the infinite temperature change rate and the finite temperature is the difficult point for the numerical calculation. If the infiniteness is improperly treated, the temperature has an infinite value, which is physically difficult in this system. Kubono removed the heat generation at that point,¹³ and Robertson did not clearly mention it.¹⁴ Kohler and Zielasek performed an analytical derivation and obtained an equation similar to equation (18).¹² However, they discussed only the temperature change at the centre of the contact region. One of the reasons may be the infinite temperature change rate.

Table 1 compares the numerical calculation [equation (20)] with the asymptotic function [equation (25)] and shows the error arising from the introduction of the asymptotic curve. The index j represents several points of . Err_j is defined as shown in Fig. 4 and is estimated as the area of the trapezoid formed by the intersection points of the two curves and the lines and . Note that when j = 1, is approximated by a rectangle. Thus, 27 Here, the temperature is calculated; thus, a precision of four significant figures in the calculation is sufficient for the entire integration range. This is the only part of the integration range; thus, we need to obtain a much greater precision, say, a precision of six significant figures. From the result shown in Table 1, we can assume that the approximation is valid within .

Table 2

Comparison of numerical calculation [equation (20)] and asymptotic function [equation (25)]

j		Equation (20)	Equation (25)	Err_j
1	10^− 10	3.59170 × 10⁴	3.59174 × 10⁴	4.2 × 10^− 11
2	10^− 9	1.13577 × 10⁴	1.13581 × 10⁴	3.8 × 10^− 10
3	10^− 8	3.59134 × 10³	3.59174 × 10³	3.7 × 10^− 9
4	10^− 7	1.13540 × 10³	1.13581 × 10³	3.6 × 10^− 8
5	10^− 6	3.58770 × 10²	3.59174 × 10²	3.6 × 10^− 7
6	10^− 5	1.13177 × 10²	1.13581 × 10²	3.6 × 10^− 6
7	10^− 4	3.55188 × 10¹	3.59174 × 10¹	3.6 × 10^− 5
8	10^− 3	1.09740 × 10¹	1.13581 × 10¹	3.5 × 10^− 4

Error estimation of asymptotic curve near

The change in the temperature distribution at the upper surface ( ) of the semi-infinite body is shown in Fig. 5a. The temperature increases as the region at the periphery of the contact area becomes larger. Then, the temperature around the centre of the contact area increases and takes almost the same value inside the contact area. Figure 5b shows the change in the temperature distribution inside the body. The temperature monotonically decreases with respect to . The temperature at decreases more gradually with respect to than at .

Change in normalised temperature : a at upper surface for several , where horizontal axis is ; b inside body at for several , where horizontal axis is ; c at upper surface and inside body, where horizontal axis is ; range of in c is expanded to d ; in b–d, solid lines represent and dotted lines represent

Figure 5a and 5b shows that, when , the temperature distribution reaches the steady state, where the amount of heat diffusion and the amount of heat generation agree at each point, and the temperature distribution does not change. From equation (16), the condition is 28 From equations (10), 14 and (28), the temperature distribution under the steady state condition is obtained as 29 This equation was first obtained by Greenwood and Williamson.⁹ The difference between the values obtained using the numerical calculation in this research and equation (29) are between ± 1.0 × 10^− 5. This is sufficient precision for the calculation of the temperature.

Figure 5c shows the temperature change at several points. When , the temperature increases at the periphery of the contact area. Then, for , the temperature at the centre of the contact area increases, taking almost the same value as at the periphery of the contact area around . Figure 5d shows the detailed temperature change for . After , the temperature almost reaches the steady state around .

Applications

We discuss the applications of the obtained results. Conventional experiments on the point contact current showed that the periphery of the contact area melted (see Fig. 5a).^{9, 11} The temperature obviously has different values at the periphery and the centre of the contact area for a while after the current begins to flow. The values become closer and have almost the same value at , as shown in Fig. 5c. The result suggests that the conventional experimental studies were done under the condition .

If the steady state condition is needed, as Fig. 5c shows, the temperature distribution almost reaches the steady state around . Thus, it is necessary to apply the current at . According to the normalisation of , the relation between t and is 30 Therefore, 31 is necessary.

To avoid melting at the contact area, it is important to know the maximum temperature. takes the maximum value . The relation between T and is 32 When the melting temperature is defined as (K), the condition at which the melting temperature is exceeded is 33 To melt the contact area, the condition opposite to that of equation (33) is necessary. Thus, 34 If V_c satisfies equation (34), there is a time to melting t_m (s). The normalised time to melting is defined by equation (30). Since the temperature takes its highest value at the periphery of the contact area, satisfies 35 We normalise V_c as ; then, equation (35) becomes 36 From Fig. 5c, and have a one to one relation. According to equation (36), and also have a one to one relation; thus, and have a one to one relation. Figure 6 shows , where satisfies from equation (34). It shows that monotonically decreases with increasing . Therefore, the time to melting is short when is large.

Relation between and

The theoretical time to melting is compared with experimental data.⁹ The electric current is passed through a contact area with a diameter of a = 5.0 × 10^-5. The voltage V_c is about 0.19. The literature shows a value of 0.38, but that is for the entire voltage drop for two semi-infinite bodies. The material is gold, so T_m = 1337.33, ρ = 2.3 × 10^-8, κ = 3.2 × 10², and D = 1.3 × 10^-4 at T₀ = 300. Thus, . From Fig. 6, . Thus, . The experiment shows that the voltage changes around t = 3.0 × 10^-5. The experimental and theoretical results are of the same order.

Regarding the melting area, when , the temperature is higher around the periphery of the contact area. Thus, the melting area is around the periphery of the contact area. The melting range is roughly estimated using Fig. 5a–5c. When , the temperature takes almost the same value in the contact area. Thus, the melting area is in the contact area. Regarding the depth range, Fig. 5b shows that the temperature at decreases more gradually with respect to than at . Thus, the melting range is deeper at the centre of the contact area than at the periphery.

Regarding the total heat generation, which corresponds to the total energy consumption, from equation (14), the heat generation in unit time δQ (J) is 37 δQ can also be derived using the relation between the electric current passing the system per unit time I (A) and V_c. From equations (5) and (11), the potential is the same over the surface, where ξ is constant, and the current density has only the component in the ξ direction. Then, if the current density is integrated over the surface, it agrees with the electric current flowing through the body per unit time. 38 where . From I and V_c, the integration of the heat generated in δt is , which corresponds to equation (37).

From equation (37), the total heat generated until melting occurs Q (J) is 39 The total heat generated Q is proportional to a³ and to . Figure 7 shows that monotonically decreases with increasing . Thus, the total heat generated Q is small when a is small and is large. Therefore, we can decrease the energy consumption by choosing a smaller a and larger .

Relation between Q and

Conclusions

The change in the temperature distribution due to constriction of an electric current through a contact area is analytically and numerically investigated. The potential distribution, current density distribution and heat density distribution are analytically obtained. They are expressed simply by introducing a spheroidal coordinate system.

The current density distribution and heat density distribution have a singular point at the periphery of the contact area. The temperature change rate with respect to time has an infinite value at because of the singularity. It is revealed that the temperature has a finite value, whereas the temperature change rate has an infinite value.

The change in the temperature distribution is shown in Fig. 5. The temperature is higher around the periphery of the contact area when . Then, the temperature around the centre of the contact area increases and takes almost the same value when . From these results, the melting area is concentrated around the periphery of the contact area when . The melting range is roughly estimated using Fig. 5a and 5b. This suggests that conventional experimental studies were done under the condition , because they reported that the periphery of the contact area melted. When , the melting area is on the contact area. The melting range is deeper at the centre of the contact area than at its periphery.

The temperature is in the steady state when . The temperature distribution exhibits good agreement with the result of Greenwood and Williamson.⁹ When , the temperature is almost in the steady state. If the steady state is needed, the condition for the steady state must be satisfied.

The maximum temperature is obtained because there is a steady state condition. Thus, the condition for not exceeding the melting temperature is obtained. The condition for melting is the opposite of the condition for avoiding melting. Under the condition for melting, there is a time to melting . The time to melting monotonically decreases with increasing , as shown in Fig. 6.

The total heat generation Q until the melting point is reached, which corresponds to the total energy consumption, is calculated. The total heat generation Q is proportional to a³ and monotonically decreases with increased . Therefore, we can decrease the energy consumption by choosing a smaller a and larger .

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