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Conduction-Diffusion in N-Dimensional settings as irreversible port-Hamiltonian systems

Luis Mora, Yann Le Gorrec, Hector Ramirez, Denis Matignon

Abstract

This work extends previous 1D irreversible port-Hamiltonian system (IPHS) formulations to boundary-controlled ND distributed parameter systems describing conduction-diffusion fluid phenomena. Within a unified and thermodynamically consistent framework, we show that conduction and diffusion can be represented through a single coherent structure that preserves global energy balance and ensures a correct characterization of entropy production. The resulting formulation provides a foundation for the systematic modeling and control of complex multi-physical processes governed by coupled transport mechanisms in N dimensions. In the longer term, this framework opens the door to structure-preserving numerical schemes capable of enforcing thermodynamic principles directly at the discretized level.

Conduction-Diffusion in N-Dimensional settings as irreversible port-Hamiltonian systems

Abstract

This work extends previous 1D irreversible port-Hamiltonian system (IPHS) formulations to boundary-controlled ND distributed parameter systems describing conduction-diffusion fluid phenomena. Within a unified and thermodynamically consistent framework, we show that conduction and diffusion can be represented through a single coherent structure that preserves global energy balance and ensures a correct characterization of entropy production. The resulting formulation provides a foundation for the systematic modeling and control of complex multi-physical processes governed by coupled transport mechanisms in N dimensions. In the longer term, this framework opens the door to structure-preserving numerical schemes capable of enforcing thermodynamic principles directly at the discretized level.
Paper Structure (9 sections, 11 theorems, 45 equations)

This paper contains 9 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. 1D irreversible port-Hamiltonian systems
  3. The 1D heat equation as an IPHS
  4. The $ND$ heat conduction as an IPHS
  5. The complete $ND$ diffusion process as an IPHS
  6. Processes with diffusion of one species only
  7. Processes with diffusion of $n$ different species
  8. The general $ND$ formulation
  9. Conclusion and Future work

Key Result

Lemma 1

The total energy balance is which leads, when the input is set to zero, to $\dot{H} = 0$ in accordance with the first law of Thermodynamics.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Lemma 1: First law of Thermodynamics
  • proof
  • Lemma 2: Second law of Thermodynamics
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3
  • proof
  • ...and 14 more