Mathematical modelling and numerical simulation of two-phase multi-component flows of CO₂ mixtures in pipes

Abstract

In this thesis, the modelling of one-dimensional two-phase flows is studied, as well as the associated numerical methods. The background for this study is the need for numerical tools to simulate fast transients in pressurised carbon dioxide pipelines, amongst other things the crack arrest problem. This is a coupled mechanical and fluid-dynamical problem, where the pressurised gas causes a crack to propagate along the pipe, while being depressurised to the atmosphere. The crack stops when the pressure at the crack tip cannot drive it any longer.

Two-phase flow models were derived from the fundamental local conservation laws for mass, momentum and total energy. Through averaging of these relations, a system of one-dimensional transport equations was obtained, that must be closed by physical models and assumptions. The underlying assumptions made in some of the classical models of the literature are made clear. The numerical methods to solve hyperbolic conservation laws, based on the Finite Volume Method, are subsequently presented.

A partially-analytical Roe scheme for the N-phase drift-flux model has been derived. The wave structure of the model is presented. It is mostly analytical, except for some thermodynamical parameters. This makes the scheme very flexible with respect to the thermodynamical relations. An algorithm to resolve the thermodynamical state of a mixture of N phases following the stiffened gas equation of state is derived.

A Roe scheme for the six-equation two-fluid model has been derived. This model needs to be regularised to be hyperbolic. A tool to verify the physical relevance of a regularisation term is provided.

The instantaneous chemical relaxation is performed on a five-equation two-fluid model to derive a four-equation model, where the phases are in full mechanical, thermal and chemical equilibrium at all times.

An application example of numerical methods to solve the crack arrest problem is presented. A method is developed to evaluate the flow through a crack, and compared in the single-phase case to an analytical method.