**Mathematical Modelling **

**(Led by UCL)**

*Work Package 1.1.1 – **Single phase equilibrium properties and phase equilibria*

- To develop, implement and validate physical property models for the prediction of equilibrium, transport properties and phase equilibria of ethylene and ethylene mixtures over a range of temperature and pressure conditions typically occurring in pipeline transportation processes and rupture.

*Work Package 1.1.2 – Transport Properties*

- Study transport properties like viscosity, Self and Maxwell-Stefan diffusion coefficients, thermal conductivity and interfacial tension to assist in accurately designing pipeline transportation and rupture outflow model.

*Work Package 1.1.3 – Parameter estimation and code implementation*

- Validation of models developed in this task against literature experimental data. Creation of a database with pure component and binary mixtures parameters. The database will also include recommendations regarding the expected accuracy of the models with respect to temperature, pressure and composition conditions
- A software code with the new models will be developed in a user-friendly framework in order to be easily integrated into the other computational tools to be generated

*Work package 1.2 – Heterogeneous flow model*

- Develop a heterogeneous flow model that will include the equations of mass, momentum and energy conservation for turbulent two-phase flow. In contrast to the conventionally used homogenous equilibrium approach, separate conservation equations will be applied to vapor and liquid phases, accounting for cross-phase mass transfer and energy exchanges. These conservation equations will apply to all flow regimes such as stratified, slug and dispersed flows.
- Estimation of the fluid/pipe wall heat transfer rate given the highly turbulent flow following pipeline rupture. Shall use phase and flow dependent fluid/wall heat transfer coefficients whilst also accounting for frictional pressure drop.
- The outflow model developed will be implemented in a computational finite-volume code for the solution of the set of hyperbolic equations for 1-D transient flows, ensuring numerical stability and ability to successfully simulate flow regime transitions.

*Work package 1.3 – In-line ESDV closure dynamic response simulation and its optimal positioning based on cost benefit analysis*

- The closure dynamics of both non-return check and ball valves will be simulated by implementing the appropriate boundary conditions into the heterogeneous flow model developed in WP 1.2
- Will employ the ESDV model developed above to develop a risk-based cost/benefit methodology
*to enable the optimal positioning of in-line ESDVs along pressurized transportation pipelines*. - The financial implications of reducing ESDV separation distance will be accounted for by calculating the cost of each valve over its lifetime based on the type of valve, the rate of depreciation and the cost of maintenance

*Work Package 1.4 – Brittle fracture model*

- The pressure and fluid temperature at the crack tip for a given starting through-wall defect size and geometry will be determined using the heterogeneous flow model developed in WP 1.2
- A transient 3D heat transfer model will next be developed to simulate the corresponding pipe-wall temperature profile in proximity to the through wall defect.
- In the case of buried pipelines where there is no blowout following a small diameter leak, we will determine the cooling rate of the surrounding soil caused by the diffusing expanding jet and the consequent drop in the pipe wall temperature in contact with it.
- To model the heat transfer between the buried pipeline and the surrounding soil, a heat transfer coefficient approximation for a horizontal cylinder in a semi-infinite medium will be employed
- If and when the crack tip falls below the DBTT, we will employ linear elastic fracture mechanics to calculate the corresponding stresses and simulate the crack propagation process