Joule Heating in Numerical Simulations

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Joule heating, also known as resistive or ohmic heating, refers to the process by which the passage of electric current through a conductor releases heat due to the electrical resistance of the material. This phenomenon is governed by Joule’s law, which states that the power dissipated as heat is directly proportional to the square of the current times the resistance of the material: $P = I^{2} R$ . In the context of numerical simulations, accurately modeling Joule heating is essential in a wide range of engineering and scientific applications including microelectronics, plasma physics, electrothermal MEMS devices, and battery design.

In finite element or finite volume simulations, Joule heating is often incorporated as a source term in the heat conduction equation. The generalized heat equation with a Joule heating source term is given by:
$ρ c_{p} \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + σ | E |^{2}$
where $ρ$ is the density, $c_{p}$ is the specific heat capacity, $T$ is the temperature, $k$ is the thermal conductivity, $σ$ is the electrical conductivity, and $E$ is the electric field. The term $σ | E |^{2}$ represents the volumetric heat generation due to Joule heating.

The electric field $E$ itself is derived from the electric potential $ϕ$ , such that $E = - \nabla ϕ$ . Hence, the source term for Joule heating becomes $σ | \nabla ϕ |^{2}$ . Accurate numerical resolution of this term requires fine spatial discretization, particularly in regions with sharp gradients in electric potential or material properties. In practice, this often demands adaptive meshing techniques and multi-physics solvers to couple the electric and thermal fields.

One of the key challenges in simulating Joule heating lies in the nonlinear coupling between temperature and electrical conductivity. In many materials, $σ$ is a function of temperature, introducing a feedback loop: as temperature increases, conductivity changes, which in turn modifies the distribution of current and thus the heat generated. This nonlinearity often necessitates iterative or fully coupled solvers to ensure convergence and stability of the simulation. In high-performance semiconductor devices, for instance, this coupling can lead to thermal runaway if not accurately captured.

Temperature Field in a Dielectrophoretic Cell Separation Device Subject to Joule Heating [1]

Moreover, in electrothermal microactuators, such as V-shaped thermal actuators used in MEMS, the device performance directly depends on the localized heating due to Joule effects. Simulations of these systems require not only the inclusion of Joule heating but also its interaction with thermal expansion and structural deformation.

In multiphysics software like COMSOL Multiphysics® or ANSYS, built-in modules allow for coupled electrothermal simulations, enabling designers to input temperature-dependent material properties and solve for the transient or steady-state temperature fields. These tools use variational formulations to solve Maxwell’s equations alongside the heat equation, providing high-fidelity results that align well with experimental data. However, the fidelity of results hinges on the accurate calibration of material models, mesh resolution, and solver tolerances.

In recent studies, the impact of Joule heating on various technological applications has been extensively analyzed through numerical simulations. For instance, researchers investigated the effects of Joule heating in electrode-based microfluidic devices, revealing that triangular electrode configurations produced less heating at applied potentials of 30 V, thereby enhancing device reliability and cell viability. 1 Similarly, a study on inductive heating in thermoplastic composites developed a numerical model considering Joule heating at the interfaces of unidirectional materials, providing insights into optimizing manufacturing processes. 2 Additionally, numerical analyses have explored the influence of Joule heating on heat transfer and entropy generation in magnetohydrodynamic (MHD) channel flows, utilizing Physics-Informed Neural Networks (PINNs) for parametric studies and addressing ill-posed problems in simulations. 3 These investigations underscore the critical role of accurately modeling Joule heating in various applications to ensure the reliability and performance of engineering systems.

Joule heating plays a pivotal role in the thermal performance and reliability of electrically driven systems. Numerical simulations that incorporate this phenomenon require careful consideration of electric field distributions, material non-linearities, and thermal boundary conditions. With the growing complexity of modern electronic and electrothermal systems, precise simulation of Joule heating is not only a necessity but also a cornerstone for robust design and innovation.

References:

Three-Dimensional Numerical Simulation of the Joule Heating

Numerical simulation of Joule heating effect

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