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A new temperature evolution equation that enforces thermodynamic vapour-liquid equilibrium in multiphase flows -- application to CO2 modeling

Pardeep Kumar, Benjamin Sanderse, Patricio I. Rosen Esquivel, R. A. W. M. Henkes

TL;DR

The paper develops a framework to model CO$_2$ depressurization by coupling the Hyperbolic Homogeneous Equilibrium Model (HEM) with UV-Flash thermodynamics under Span-Wagner EOS. It introduces two reduced VLE strategies: Reduced-VLE-Algebraic, which collapses the four VLE equations to a single algebraic one using saturation data, and Reduced-VLE-ODE, which differentiates the reduced equation to yield a temperature-evolution ODE for explicit time integration. These approaches are demonstrated on tank and pipeline depressurization problems, achieving substantial computational speedups with acceptable accuracy and highlighting energy-conservation trade-offs in the ODE variant. The results validate the methods against Hammer benchmarks and reveal the potential for efficient CO$_2$ transport simulations, while outlining avenues for extension to multi-component systems and improved time integration.

Abstract

This work presents a novel framework for numerically simulating the depressurization of tanks and pipelines containing carbon dioxide (CO2). The framework focuses on efficient solution strategies for the coupled system of fluid flow equations and thermodynamic constraints. A key contribution lies in proposing a new set of equations for phase equilibrium calculations which simplifies the traditional vapor-liquid equilibrium (VLE) calculations for two-phase CO2 mixtures. The first major novelty resides in the reduction of the conventional four-equation VLE system to a single equation, enabling efficient solution using a non-linear solver. This significantly reduces computational cost compared to traditional methods. Furthermore, a second novelty is introduced by deriving an ordinary differential equation (ODE) directly from the UV-Flash equation. This ODE can be integrated alongside the governing fluid flow equations, offering a computationally efficient approach for simulating depressurization processes.

A new temperature evolution equation that enforces thermodynamic vapour-liquid equilibrium in multiphase flows -- application to CO2 modeling

TL;DR

The paper develops a framework to model CO depressurization by coupling the Hyperbolic Homogeneous Equilibrium Model (HEM) with UV-Flash thermodynamics under Span-Wagner EOS. It introduces two reduced VLE strategies: Reduced-VLE-Algebraic, which collapses the four VLE equations to a single algebraic one using saturation data, and Reduced-VLE-ODE, which differentiates the reduced equation to yield a temperature-evolution ODE for explicit time integration. These approaches are demonstrated on tank and pipeline depressurization problems, achieving substantial computational speedups with acceptable accuracy and highlighting energy-conservation trade-offs in the ODE variant. The results validate the methods against Hammer benchmarks and reveal the potential for efficient CO transport simulations, while outlining avenues for extension to multi-component systems and improved time integration.

Abstract

This work presents a novel framework for numerically simulating the depressurization of tanks and pipelines containing carbon dioxide (CO2). The framework focuses on efficient solution strategies for the coupled system of fluid flow equations and thermodynamic constraints. A key contribution lies in proposing a new set of equations for phase equilibrium calculations which simplifies the traditional vapor-liquid equilibrium (VLE) calculations for two-phase CO2 mixtures. The first major novelty resides in the reduction of the conventional four-equation VLE system to a single equation, enabling efficient solution using a non-linear solver. This significantly reduces computational cost compared to traditional methods. Furthermore, a second novelty is introduced by deriving an ordinary differential equation (ODE) directly from the UV-Flash equation. This ODE can be integrated alongside the governing fluid flow equations, offering a computationally efficient approach for simulating depressurization processes.
Paper Structure (31 sections, 49 equations, 15 figures, 4 tables)

This paper contains 31 sections, 49 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Fluid flow in a pipeline
  4. Tank model
  5. Thermodynamics and standard UV-Flash overview
  6. New vapour-liquid equilibrium equations
  7. Reduced algebraic vapour-liquid equilibrium equation
  8. Reduced differential vapour-liquid equilibrium equation
  9. Summary of equations for VLE
  10. Coupling the ODE formulation to the tank and pipe equations
  11. Time integration methods
  12. Algebraic approaches: DAE interpretation
  13. ODE approach
  14. Transition from single-phase to two-phase
  15. Numerical experiments
  16. ...and 16 more sections

Figures (15)

  • Figure 1: Saturation curve/space for $\mathrm{CO_2}$ in $p-T$ and $\rho-e$ co-ordinates
  • Figure 2: Comparison of tank depressurization results with Hammer et al. $\Delta t = 1$ s.
  • Figure 3: Simulation trajectory for the Hammer et al. test case, $\Delta t = 1$ s.
  • Figure 4: Approximation error from Full-VLE-Algebraic to Reduced-VLE-Algebraic, $\Delta t = 1$ s.
  • Figure 5: Constraint error from Reduced-VLE-Algebraic to Reduced-VLE-ODE, $\Delta t = 1$ s.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2