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Mathematical modelling and numerical simulation of two-phase multi-component flows of CO₂ mixtures in pipes
Norwegian University of Science and Technology, Faculty of Engineering Science and Technology, Department of Energy and Process Engineering.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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.

NTNU, 2012.
##### Series
Doctoral theses at NTNU, ISSN 1503-8181 ; 2012:296
##### National Category
Machinery energy and environmental technology
##### Identifiers
ISBN: 978-82-471-3907-3,hISBN: 978-82-471-3908-0, electronicOAI: oai:DiVA.org:ntnu-19743DiVA: diva2:580253
##### Public defence
2012-12-14, 00:00
Available from: 2013-01-09 Created: 2012-12-21 Last updated: 2013-01-09Bibliographically approved
##### List of papers
1. Numerical Resolution of CO2 Transport Dynamics
Open this publication in new window or tab >>Numerical Resolution of CO2 Transport Dynamics
2009 (English)In: Proceedings of the SIAM Mathematics for industry ’09 conference, 2009Conference paper (Refereed)
##### Identifiers
urn:nbn:no:ntnu:diva-19738 (URN)
##### Conference
Mathematics for industry '09
Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2013-01-09Bibliographically approved
2. On solutions to equilibrium problems for systems of stiffened gases
Open this publication in new window or tab >>On solutions to equilibrium problems for systems of stiffened gases
2011 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 71, no 1, 41-67 p.Article in journal (Refereed) Published
##### Identifiers
urn:nbn:no:ntnu:diva-19740 (URN)10.1137/100784321 (DOI)
##### Note

Available from: 2012-12-21 Created: 2012-12-21 Last updated: 2013-01-09Bibliographically approved
3. Wave propagation in multicomponent flow models
Open this publication in new window or tab >>Wave propagation in multicomponent flow models
2010 (English)In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 70, no 8, 2861-2882 p.Article in journal (Refereed) Published
##### Identifiers
urn:nbn:no:ntnu:diva-19741 (URN)10.1137/090777700 (DOI)
##### Note

Available from: 2012-12-21 Created: 2012-12-21 Last updated: 2013-01-09Bibliographically approved
4. Towards a formally path-consistent Roe scheme for the six-equation two-fluid model
Open this publication in new window or tab >>Towards a formally path-consistent Roe scheme for the six-equation two-fluid model
2010 (English)In: AIP Conference Proceedings, ISSN 0094-243X, Vol. 1281, no 1, 71-74 p.Article in journal (Refereed) Published
##### Identifiers
urn:nbn:no:ntnu:diva-19734 (URN)10.1063/1.3498580 (DOI)
##### Note

Copyright 2012 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

The following article appeared in ICNAAM 2010: International Conference of Numerical Analysis and Applied Mathematics 2010 and may be found at  http://dx.doi.org/10.1063/1.3498580

Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2013-01-09Bibliographically approved
5. A Roe scheme for a compressible six-equation two-fluid model
Open this publication in new window or tab >>A Roe scheme for a compressible six-equation two-fluid model
2012 (English)In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 70, no 11Article in journal (Refereed) Published
##### Identifiers
urn:nbn:no:ntnu:diva-19735 (URN)10.1002/fld.3752 (DOI)
##### Note

Morin, A., Flåtten, T. and Munkejord, S. T. (2012), A Roe scheme for a compressible six-equation two-fluid model. Int. J. Numer. Meth. Fluids. doi: 10.1002/fld.3752

Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2013-01-09Bibliographically approved
6. On interface transfer terms in two-fluid models
Open this publication in new window or tab >>On interface transfer terms in two-fluid models
2012 (English)In: International Journal of Multiphase Flow, ISSN 0301-9322, Vol. 45, 24-29 p.Article in journal (Refereed) Published
##### Abstract [en]

In this note we consider two-fluid models based on the usual formulations for conservation of mass, total momentum and total energy. We present some potentially useful general relationships between the interface exchange terms and the evolution of the mechanical variables. In particular, we discuss the possibility of obtaining in this framework a model that is both thermodynamically reversible and possesses real eigenvalues. We formally prove that such a model must include terms associated with the virtual mass force.We then address a technical issue regarding the modelling of interface transfer terms in the energy equations. In particular, we demonstrate how the formulation of the non-conservative products in these equations determine whether the interface exchange terms represent heat or energy transfer. © 2012 Elsevier Ltd.

##### Identifiers
urn:nbn:no:ntnu:diva-19736 (URN)10.1016/j.ijmultiphaseflow.2012.05.002 (DOI)
##### Note

The article is reprinted with kind permission from Elsevier, sciencedirect.com

Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2013-01-09Bibliographically approved
7. A Two-Fluid Four-Equation Model with Instantaneous Thermodynamical Equilibrium
Open this publication in new window or tab >>A Two-Fluid Four-Equation Model with Instantaneous Thermodynamical Equilibrium
##### Identifiers
urn:nbn:no:ntnu:diva-19742 (URN)
Available from: 2012-12-21 Created: 2012-12-21 Last updated: 2013-01-09Bibliographically approved
8. Pipeline Flow Modelling with Source Terms Due to Leakage: The Straw Method
Open this publication in new window or tab >>Pipeline Flow Modelling with Source Terms Due to Leakage: The Straw Method
2012 (English)In: Energy Procedia, ISSN 1876-6102, Vol. 23, 226-235 p.Article in journal (Refereed) Published
##### Abstract [en]

A numerical method for calculating the leakage flow rate through a crack in a pressurized pipeline is presented. Calculations of the flow inside, and leakage from, a pipeline with a running crack are performed. For the case of single-phase flow, the flow through the crack can also be calculated using choked-flow theory. The two methods are compared and identical results obtained. The advantage of the present method is that it does not rely on analytical expressions for the flow through the crack, and it is therefore thought to be applicable for two-phase flow, including phase transition. Hence it may be of use in the development of coupled fluid-structure models for the assessment of running ductile fracture in carbon dioxide transport pipelines. Copyright © 2012 Published by Elsevier Ltd.

##### Identifiers
urn:nbn:no:ntnu:diva-19737 (URN)10.1016/j.egypro.2012.06.023 (DOI)
##### Note

NOTICE: this is the author’s version of a work that was accepted for publication in Energy Procedia. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Energy Procedia, VOL 23, DOI:10.1016/j.egypro.2012.06.023

Available from: 2012-12-20 Created: 2012-12-20 Last updated: 2013-01-09Bibliographically approved

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##### By organisation
Department of Energy and Process Engineering
##### On the subject
Machinery energy and environmental technology