# REE13: Mathematical modelling of erosion and corrosion in water-cooled systems

Researcher: | Dr Anthony Lock |

Team Leader(s): | Dr Ian Griffiths |

Collaborators: | Prof. Howard Stone, Princeton |

Dr Daniele Vigolo, Princeton | |

Phil Heitzenroeder, Princeton |

Project completed May 31, 2011

**Background**

The ITER project aims to show that it is possible to produce commercial energy from fusion (the joining of atomic nuclei). If realised, such an energy source would help to meet increasing global demand and would also help to address the environmental concerns associated with current forms of energy production.

To achieve fusion, hot plasma will be magnetically contained within an 11-metre high doughnut-shaped vessel called a tokamak. Cooling pipes that circulate water through the system, called Edge-Localized-Mode (ELM) coils, will play a crucial role in transporting heat away from the tokamak. The ELM coils will be worn over time due to corrosion and erosion of the pipes from the flowing water. Since maintenance and repair of the ELM coils will require a complete shut-down of operations, the coils should be designed to have a lifetime of 20 years.

To assist in the design of the ELM coils, the researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) mathematically modelled corrosion and erosion of the pipes, taking into consideration filtering of the resulting particulates in the water flow to reduce further erosion.

For the model, pipe wear is assumed to be caused by corrosion and erosion. Corrosion of the pipes releases particulates into the water. Circulating particulates collide with the pipe’s interior causing erosion. Since water in the ELM coils will be recirculated through a common water volume, a key concern is that erosion might lead to exponential growth in the number of particulates. As more particulates are released into the water flow by erosion, this accelerates the amount of erosion, leading to even more particulates in the water.

To mitigate the accumulation of particulates in the water, the proposed system design diverts one percent of the total volume of flow for filtering, and then it re-enters the main flow. It is assumed that all particles above a threshold size (typically 10µm) are captured by the filter.

**Techniques and Challenges**

The Finnie model for the erosion of a ductile material is used. In this model, the volume of material eroded from the interior of the pipe depends on the mass, speed and angle of impact of the particulates in the flow. Thus, it also depends on turbulence and the path of the pipe. Specifically, erosion consists of turbulent effects throughout the pipe and also inertial effects at pipe bends. The products of corrosion are assumed to enter the flow at a constant rate. The model yields a first-order linear ordinary differential equation for the evolution of the total mass of particulates in the system.

In the proposed design, particles are captured using cross-flow filtering. Filtering is modelled by considering a T-shaped set-up consisting of a secondary channel that branches perpendicularly from a larger main pipe as shown in Figure 2. By assuming two-dimensional, steady, laminar flow, a particle’s trajectory can be numerically calculated using Newton’s second law for a spherical particle subject to fluid drag. A particle is assumed to be filtered if its trajectory enters the secondary channel. The time evolution of particulates in the system depends crucially upon the fraction of entrained particles that reach the filter.

**Results**

The developed model was able to provide an assessment of the current design for the ELM coils regarding erosion and the proposed filtering scheme.

A key dimensionless parameter is the Stokes number which provides a measure of how closely particles follow the streamlines. In this model, for the particle sizes of interest (10µm to 1mm), the Stokes number was found to be small. Thus, particles closely follow the streamlines and the cross-filter design should be effective in capturing particles in this range.

Estimates show that filtering dominates erosion leading to the prediction of a steady state in the number of particles However, a steady state does not mean that there is no longer erosion, rather that exponential growth in the number of particulates will not occur.