Analytical Modeling of Joule Heating in Electro-Thermal Contacts for Short-Term Industrial Applications

At short times, thermal disturbances applied to the boundaries only gradually penetrate the medium. Thus, at a relatively short time after the beginning of the phenomenon, the disturbance only reaches a layer of thickness 'e', which is called the penetration depth. Beyond this penetration depth, the rest of the medium has not yet participated in the transient process and is still at its initial condition. Therefore, at this short time interval, the length of the medium has no influence on the behavior of temperature. The assumption of semi-infinite media consists in considering that the length of the medium is infinite.

The chosen model for the electrothermal contact problem involves placing two cylindrical bars of equal circular cross-section in (imperfect) contact across the entirety of their base areas, as shown in Figure 1. The two cylinders are assumed to be of equal size (L₁=L₂ and D₁=D₂). This model involves a study in the transient stat, where the imperfect contact creates a thermal contact resistance Rtc. When an electric current passes through the contact, the imperfection of the contact also creates an electrical contact resistance REc. With this passage of electric current, other source terms will appear, such as the heat flux generated at the interface φg(W/m²) due to the imperfect contact, and the volumetric powers dissipated in the two cylinders P₁ and P₂(W/m²). it should be noted that the numerical values used for solving the problem in this study are inspired by the work of Le Meur et al. [13] and have been modified to fit the specific requirements of the electro-thermal contact problem being investigated. Additionally, two temperatures TL₁=0℃ and TL₂=0℃ are imposed at the left and right boundaries, respectively, and the lateral surface of the two cylinders is assumed to be perfectly insulated.

1.png

Figure 1. Schematic illustration of the electro-thermal contact model

The literature review [13-16, 18, 19] shows that previous studies have mainly focused on the long-time problem of solid-to-solid thermal contacts, whether for the direct problem or for the estimation and modeling of the parameters of the contact interface with heat generation. They have all investigated the thermal aspect of the macroscopic scale in long-time regimes, but none have looked into the short-time electro-thermal contact problem. This gap in the literature highlights the importance of our study, which aims to investigate the short-time electro-thermal contact problem to contribute to a better understanding of the physical phenomena that occur at the contact interface in these situations.

The model that describes the behavior of electrothermal contact in transient state is presented as follows:

The heat equation in area 1 and 2

$\left\{\begin{array}{rr}\frac{\partial^2 T_1}{\partial x^2}+\frac{P_1}{\lambda_1}=\frac{1}{a_1} \frac{\partial T_1}{\partial t} & x \in\left[-L_1, 0\right] \\ \frac{\partial^2 T_2}{\partial x^2}+\frac{P_2}{\lambda_2}=\frac{1}{a_2} \frac{\partial T_2}{\partial t} & x \in\left[0, L_2\right]\end{array}\right.$     (1)

Condition at the border

$\mathrm{x}=-\mathrm{L} 1: \; T_1\left(-L_1, t\right)=T_{-L 1}(t)$     (2)

Conservation of the flow at the interface

$\lambda_1 \frac{\partial T_1(0, t)}{\partial x}=\lambda_2 \frac{\partial T_2(0, t)}{\partial x}+\varphi_g$     (3)

Fourier condition

$\lambda_2 \frac{\partial T_2(0, t)}{\partial x}+\alpha \varphi_g=\frac{T_2(0, t)-T_1(0, t)}{R_{t c}}$     (4)

Boundary condition at the frontier

$\mathrm{x}=-\mathrm{L} 1: \; T_2\left(L_2, t\right)=T_{L_2}(t)$     (5)

The initial condition in area 1 and 2 

$\left\{\begin{array}{l}T_1(x, 0)=T_{i_1}(x) \\ T_2(x, 0)=T_{i_2}(x)\end{array}\right.$     (6)

2.1 Application of the transformation to equations

As mentioned above, disturbances during short times primarily impact small thicknesses, justifying the consideration of infinite lengths. However, the system diverges from this norm as the heating associated with volumetric power dissipation extends along the entire length of both cylinders right from the initiation of the phenomenon.

To address this unique characteristic, a variable change defined by Eq. (7) is strategically employed:

$T_j=T P_j+T_{\varphi j} j=1,2$     (7)

The transformation introduced in the context of the electrothermal contact problem serves to decompose the temperature (T) profiles into two distinct components:

where,

$T P_j$ (Volumetric Power Dissipation):

• Captures the temperature increase due to the volumetric powers dissipated in the cylinders during the transient state.
• Associated with the effects of the electric current passing through the contact, resulting in heat generation within the medium.

$T_{\phi j}$ (Generated Flux):

• Represents the temperature increase due to the generated flux at the interface.
• Accounts for the heat flux generated at the imperfect contact interface, contributing to the overall thermal behavior of the system.

Applying the variable change Eq. (7) to the original PDE Eq. (1) yields two transformed expressions:

$\left\{\begin{array}{c}\frac{\partial^2 T_{\varphi j}}{\partial x^2}=\frac{1}{a_j} \frac{\partial T_{\varphi j}}{\partial t} \\ \frac{\partial^2 T_{P j}}{\partial x^2}+\frac{P_j}{\lambda_j}=\frac{1}{a_j} \frac{\partial T_{P j}}{\partial t}\end{array}\right.$     (8)

These transformed equations separate the original PDE into two components: T_ϕj, related to the generated flux, and TP_j related to the volumetric powers.

2.2 System decomposition

The entire transient state 'A' yields two systems: 'B' and 'C'

• 'B' representing the heating due to the volumetric powers dissipated in the two cylinders during the transient state, which will be solved over finite lengths;
• 'C' representing the heating due to the generated flux during the transient stat, which will be solved by considering that both lengths L₁ and L₂ tend towards infinity.
• Case 1: Heating due to volumetric powers (finite lengths)

2.3 System 'B' - volumetric power dissipation (finite lengths)

System 'B' is designed to analyze the electrothermal contact problem when considering the heating effect due to volumetric power dissipation. The focus is on finite lengths, providing a detailed understanding of temperature profiles in this scenario.

1. Equations for finite lengths

$\begin{cases}\frac{\partial^2 T_{P 1}}{\partial x^2}+\frac{P_1}{\lambda_1}=\frac{1}{a_1} \frac{\partial T_{P 1}}{\partial t} & x \in[-\infty, 0] \\ \frac{\partial^2 T_{P 2}}{\partial x^2}+\frac{P_2}{\lambda_2}=\frac{1}{a_2} \frac{\partial T_{P 2}}{\partial t} & x \in[0,+\infty]\end{cases}$     (9)

2. Boundary and interface conditions

$T_{P 1}\left(-L_1, t\right)=0$     (10)

$\lambda_1 \frac{\partial T_{P 1}(0, t)}{\partial x}=\lambda_2 \frac{\partial T_{P 2}(0, t)}{\partial x}$     (11)

$\lambda_2 \frac{\partial T_{P 2}(0, t)}{\partial x}=\frac{T_{P 2}(0, t)-T_{P 1}(0, t)}{R_{t c}}$     (12)

$T_{P 2}\left(+L_2, t\right)=0$     (13)

$\left\{\begin{array}{c}T_{P 1}(x, 0)=0 \\ T_{P 2}(x, 0)=0\end{array}\right.$     (14)

3. Solution strategy

The selection of Laplace transforms for solving heat transfer problems in the temporal domain is grounded in the nature of heat conduction equations commonly used in various systems such as welding, non-contact interactions, and more. Heat conduction problems are often formulated as partial differential equations with appropriate boundary conditions that describe the behavior of the system. Laplace transforms have demonstrated significant efficacy in handling linear differential equations, making them particularly well-suited for problems involving heat transfer.

The rationale behind transforming the system from the temporal domain to the Laplace domain lies in the inherent advantages of Laplace transforms in simplifying the mathematical expressions associated with heat conduction. The conversion allows for a more straightforward and systematic analysis of the problem, providing a clear representation of the system's response to electrothermal interactions.

System B undergoes transformation through the application of the Laplace transform, allowing the expression of equations in the Laplace domain. This transformative process significantly simplifies the analysis of transient behavior, resulting in the following solution when applied to Eq. (9):

$\frac{\partial^2 T_{P j}}{\partial x^2}-\frac{1}{a_j} \frac{\partial T_{P j}}{\partial t}=-\frac{P_j}{\lambda_j} \rightarrow \frac{\partial^2 \theta_{P j}}{\partial x^2}-\frac{1}{a_j}\left[s . \theta_{P j}-T_{P j}(x, 0)\right]=-\frac{P_j}{s . \lambda_j}$

With T_P(x, 0)=0 and 's' the Laplace variable.

$\left\{\begin{array}{l}\frac{\partial^2 \theta_{P 1}}{\partial x^2}-\frac{s}{a_1} \theta_{P 1}=-\frac{P_1}{\lambda_1} \cdot s^{-1} \\ \frac{\partial^2 \theta_{P 2}}{\partial x^2}-\frac{s}{a_2} \theta_{P 2}=-\frac{P_2}{\lambda_2} \cdot s^{-1}\end{array}\right.$     (15)

The solutions for the transformed Eq. (15) are detailed as follows:

$\left\{\begin{array}{l}\theta_{P 1}(x, s)=C_1 e^{-q_1 x}+C_2 e^{+q_1 x}+\frac{a_1 P_1}{\lambda_1} \cdot s^{-2} \\ \theta_{P 2}(x, s)=C_3 e^{-q_2 x}+C_4 e^{+q_2 x}+\frac{a_2 P_2}{\lambda_2} \cdot s^{-2}\end{array}\right.$     (16)

4. Numerical inversion

The Stehfest method [20] is employed for numerical inversion in the Laplace domain due to the unconventional nature of the temperature field solution. The solution obtained in the Laplace domain doesn't conform to standard functions, making direct inversion using traditional methods challenging. The Stehfest method is chosen for its proven effectiveness in handling such non-standard functions, offering a reliable and efficient approach to invert the Laplace-transformed equations. Its application ensures accurate insights into the temporal evolution of the electrothermal contact system, addressing the specific complexities associated with the analytical framework utilized in this study.

The solutions for the transformed Eq. (16) involve coefficients C₁, C_2, C₃ and C₄, which are determined by applying boundary conditions. These coefficients play a crucial role in shaping the behavior of the system during the transient state.

To execute the inversion from θ_P1(x,s) → T_P1(x,t), the Stehfest numerical inversion method is applied. This involves calculating T_Pj(x,t) using the formula:

$T_{P j}(x, t)=\left(\frac{\ln (2)}{t}\right) \sum_{j=1}^N V_j(N) \theta_{P j}\left(j \frac{\ln (2)}{t}\right)$     (17)

The term V_j is computed using the given formula:

$V_j=(-1)^{\frac{N}{2}+j} \sum_{k=\left|\frac{j+1}{2}\right|}^{\min \left(i, \frac{N}{2}\right)} \frac{k^{\frac{N}{2}}(2 k) !}{\left(\frac{N}{2}-k\right) ! k !(k-1) !(j-k) !(2 k-j) !}$     (18)

Setting N=10 in the Stehfest numerical inversion method strikes a balance between computational efficiency and precision. This choice, representing single precision, is a standard compromise for numerical simulations, offering satisfactory results for various engineering applications. It reflects a pragmatic decision, acknowledging the diminishing returns in accuracy with higher N values while ensuring manageable computational demands for the electrothermal contact problem under consideration.

• Case 2: Heating due to the generated flow

2.4 System 'C' - generated flow (infinite lengths)

The system 'C' of equations is established to describe the thermal behavior in scenarios where the generated flow influences the entire length. The partial differential equations (PDEs) under consideration are expressed in terms of temperature profiles and their spatial and temporal derivatives.

1. Equations for infinite lengths

$\begin{cases}\frac{\partial^2 T_{\varphi 1}}{\partial x^2}=\frac{1}{a_1} \frac{\partial T_{\varphi 1}}{\partial t} & x \in[-\infty, 0] \\ \frac{\partial^2 T_{\varphi 2}}{\partial x^2}=\frac{1}{a_2} \frac{\partial T_{\varphi 2}}{\partial t} & x \in[0,+\infty]\end{cases}$     (19)

2. Boundary and interface conditions

The set of conditions for System 'C' includes initial and boundary conditions that ensure the physical relevance of the solution.

$T_{\varphi 1}(x \rightarrow-\infty, t)=0$     (20)

$\lambda_1 \frac{\partial T_{\varphi 1}(0, t)}{\partial x}=\lambda_2 \frac{\partial T_{\varphi 2}(0, t)}{\partial x}+\varphi_g$     (21)

$\lambda_2 \frac{\partial T_{\varphi 2}(0, t)}{\partial x}+\alpha \varphi_g=\frac{T_{\varphi 2}(0, t)-T_{\varphi 1}(0, t)}{R_{t c}}$     (22)

$T_{\varphi 2}(x \rightarrow+\infty, t)=0$     (23)

$\left\{\begin{array}{l}T_{\varphi 1}(x, 0)=0 \\ T_{\varphi 2}(x, 0)=0\end{array}\right.$     (24)

3. Solution strategy

By applying the Laplace transform to Eq. (19), the resulting expression is:

$\rightarrow \frac{\partial^2 \Theta_{\varphi 1}}{\partial x^2}-\frac{s}{a_1} \Theta_{\varphi 1}(x, s)=0 \rightarrow \Theta_{\varphi 1}(x, s)$$=C_1 e^{-q_1 x}+C_2 e^{+q_1 x}$ With $q_1=\sqrt{\frac{s}{a_1}}$

$\rightarrow \frac{\partial^2 \Theta_{\varphi 2}}{\partial x^2}-\frac{s}{a_2} \Theta_{\varphi 2}(x, s)=0 \rightarrow \Theta_{\varphi 2}(x, s)$$=C_3 e^{-q_2 x}+C_4 e^{+q_2 x}$ With $q_2=\sqrt{\frac{s}{a_2}}$

When x tends to -∞ the temperature reaches a finite value which implies that C₁=0.

When x tends to +∞ the temperature reaches a finite value which implies that C₄=0.

$\left\{ \begin{matrix}   \begin{align}  & {{\Theta }_{P1}}(x,s)={{A}_{1}}{{e}^{+{{q}_{1}}x}} \\ & \text{              }\centerdot  \\\end{align}  \\   {{\Theta }_{P2}}(x,s)={{A}_{2}}{{e}^{-{{q}_{2}}x}}  \\\end{matrix} \right.$     (25)

4. Numerical inversion

The coefficients (A₁ and A₂) are determined by applying boundary conditions to ensure a physically meaningful solution.

$A_2=\frac{\varphi_g\left(E_1 R_{t c} \alpha \cdot s^{-\frac{1}{2}}+1\right)}{R_{t c} E_1 E_2 \cdot s+\left(E_1+E_2\right) \cdot s^{-\frac{1}{2}}} \cdot s^{-1}$     (26)

$A_1=\frac{\varphi_g \cdot s^{-2} \cdot\left(\alpha R_{t c}^2 E_1 E_2+E_1 R_{t c} \alpha+R_{t c} E_2\right)+\varphi_g \cdot s^{-1} \cdot\left(1-R_{t c} \alpha\right)}{R_{t c} E_1 E_2 \cdot s+\left(E_1+E_2\right) \cdot s^{-\frac{1}{2}}}$     (27)

With:

E: thermal effusivity $E_j=\sqrt{ } \lambda_j c_j \rho_j$

Numerical inversion was used by the Stehfest method, as mentioned earlier since there was no way for an analytic inversion.

2.5 System 'D' - Generated Flow (Finite Lengths)

System 'D' extends the analysis of the electrothermal contact problem in the presence of the generated flow, focusing on finite lengths. It provides insights into the system's behavior when considering the influence of the generated flux within specific length constraints.

1. Equations for infinite lengths

$\begin{cases}\frac{\partial^2 T_{\varphi 1}}{\partial x^2}=\frac{1}{a_1} \frac{\partial T_{\varphi 1}}{\partial t} & x \in\left[-L_1, 0\right] \\ \frac{\partial^2 T_{\varphi 2}}{\partial x^2}=\frac{1}{a_2} \frac{\partial T_{\varphi 2}}{\partial t} & x \in\left[0, L_2\right]\end{cases}$     (28)

2. Boundary and interface conditions

$T_{\varphi 1}\left(-L_1, t\right)=0$     (29)

$\lambda_1 \frac{\partial T_{\varphi 1}(0, t)}{\partial x}=\lambda_2 \frac{\partial T_{\varphi 2}(0, t)}{\partial x}+\varphi_g$     (30)

$\lambda_2 \frac{\partial T_{\varphi 2}(0, t)}{\partial x}+\alpha \varphi_g=\frac{T_{\varphi 2}(0, t)-T_{\varphi 1}(0, t)}{R_{t c}}$     (31)

$T_{\varphi 2}\left(L_2, t\right)=0$     (32)

$\left\{\begin{array}{l}T_{\varphi 1}(x, 0)=0 \\ T_{\varphi 2}(x, 0)=0\end{array}\right.$     (33)

3. Solution strategy

By applying the Laplace transform to Eq. (28), the transformed equation is obtained as follows:

$\rightarrow \frac{\partial^2 \theta_{\varphi 1}}{\partial x^2}-\frac{s}{a_1} \theta_{\varphi 1}(x, s)=0 \rightarrow \theta_{\varphi 1}(x, s)$$=A_1 e^{q_1 x}+A_2 e^{-q_1 x}$ With $q_1=\sqrt{\frac{s}{a_1}}$

$\rightarrow \frac{\partial^2 \Theta_{\varphi 2}}{\partial x^2}-\frac{s}{a_2} \Theta_{\varphi 2}(x, s)=0 \rightarrow \Theta_{\varphi 2}(x, s)$$=B_1 e^{q_2 x}+B_2 e^{-q_2 x}$ With $q_2=\sqrt{\frac{s}{a_2}}$

4. Numerical inversion

The Ai and Bi coefficients are found by applying the boundary conditions, solving the following matrix system:

$\left(\begin{array}{cccc}e^{-q_1 L_1} & e^{q_1 L_1} & 0 & 0 \\ 0 & 0 & e^{q_2 L_2} & e-q_2 L_2 \\ q_1 \lambda_1 & -q_1 \lambda_1 & -q_2 \lambda_2 & q_2 \lambda_2 \\ 1 & 1 & \left(q_2 \lambda_2 R_{t c}-1\right) & -\left(q_2 \lambda_2 R_{t c}+1\right)\end{array}\right) \times\left(\begin{array}{c}A_1 \\ A_2 \\ B_1 \\ B_2\end{array}\right)$$=\left(\begin{array}{c}T_{L 1} \\ T_{L 2} \\ \frac{\varphi_g}{s} \\ -\alpha \varphi_g R_{t c} \boldsymbol{s}^{-\mathbf{1}}\end{array}\right)$

The analytical framework employed plays a pivotal role in elucidating the temperature field or distribution during electrothermal contact. This, in turn, facilitates the assessment of key parameters that govern heat transfer in such contact scenarios. The results derived from the analytical model, manifesting as temperature profiles or distributions, offer a comprehensive depiction of the thermal behavior within the system.

A detailed analysis of these results provides valuable insights into the distribution, propagation, and dissipation of heat during electrothermal contact. The temperature profiles allow for a nuanced understanding of transient behavior, showcasing variations at different time intervals and positions within the contact interface.

Furthermore, interpreting these results contributes to a better understanding of how various factors, such as material properties, contact resistance, and generated heat flux, impact the overall thermal performance. This interpretation enables the identification of patterns, critical points, and an evaluation of the effectiveness of electrothermal contact under different conditions.

Electrothermal phenomena play a crucial role in numerous industrial applications, where the interaction between electricity and heat transfer is of paramount importance. The study of electrothermal processes aims to understand the complex interplay between electrical currents and thermal effects, enabling the development of efficient and reliable systems.

This work investigates the problem of electrothermal contact, with a focus on heat conduction during short time intervals. Specifically, it develops an analytical approach to solve the direct problem of heat conduction in electrothermal contacts, taking into consideration the dissipation of heat due to Joule heating. This aspect has received limited attention in the existing literature, indicating the need for further investigation.

Additionally, a semi-analytical model is employed, combining analytical techniques with numerical methods to provide a robust and accurate description of the heat transfer process. The model is validated and tested using the Comsol software, which enables comprehensive simulations and analysis of electrothermal contact phenomena.

• The investigation of electrothermal contact phenomena, focusing on short time intervals, has provided valuable insights for industrial applications such as electric resistance spot welding.
• The semi-analytical method used in this study shows convergence at short times for more diffuse materials like Titanium and Aluminum.
• The depth of penetration and temperature changes can be accurately predicted within a time frame of 6 seconds for Aluminum.
• For Titanium, the convergence of the semi-analytical method occurs at longer times, around 33 seconds, due to differences in thermal diffusivity.
• Considering finite lengths in the modeling approach provides additional insights into the heat transfer process.
• The model converges for both short and long times, regardless of the material's diffusivity, when finite lengths are considered.
• The implications of these findings are significant for electric resistance spot welding, as engineers can optimize welding parameters for both Titanium and Aluminum.
• Understanding the different thermal behaviors of these materials allows for precise control over the welding process.
• The limitations of the semi-infinite assumption in modeling have been identified, emphasizing the importance of considering finite lengths for accurate temperature predictions.

The validated semi-analytical model presents potential as a direct tool for determining key parameters governing electrothermal contacts. It can play a crucial role in experimental studies aimed at identifying and characterizing critical factors such as the partition coefficient, thermal resistance, and electrical resistance in electrothermal contacts. This potential application opens avenues for more targeted and informed experimental investigations, fostering a deeper understanding of underlying mechanisms and providing a foundation for optimizing the design and performance of electrothermal contact devices and systems. In the future, research could focus on utilizing the validated semi-analytical model to extract practical parameters, bridging the gap between theoretical knowledge and real-world applications. This approach has the potential to enhance the accuracy and applicability of electrothermal contact modeling in various industrial contexts.