A large amount of nuclear waste is stored in tailings ponds as a solid-liquid slurry, and liquid flows containing suspensions of solid particles are encountered in the treatment and disposal of this waste. In processing this waste, it is important to understand the behaviour of particles within the flow in terms of their settling characteristics, their propensity to form solid beds, and the re-suspension characteristics of particles from a bed. A clearer understanding of such behaviour would allow the refinement of current approaches to waste management, potentially leading to reduced uncertainties in radiological impact assessments, smaller waste volumes and lower costs, accelerated clean-up, reduced worker doses, enhanced public confidence and diminished grounds for objection to waste disposal. Mathematical models are of significant value in nuclear waste processing since the extent of characterisation of wastes is in general low. Additionally, waste processing involves a diverse range of flows, within vessels, ponds and pipes. To investigate experimentally all waste form characteristics and potential flows of interest would be prohibitively expensive, whereas the use of mathematical models can help to focus experimental studies through the more efficient use of existing data, the identification of data requirements, and a reduction in the need for process optimisation in full-scale experimental trials. Validated models can also be used to predict waste transport behaviour to enable cost effective process design and continued operation, to provide input to process selection, and to allow the prediction of operational boundaries that account for the different types and compositions of particulate wastes. In this paper two mathematical modelling techniques, namely Reynolds-averaged Navier-Stokes (RANS) and large eddy simulation (LES), have been used to investigate particle-laden flows in a straight square duct and a duct with a bend. The flow solutions provided by these methods have been coupled to a three-dimensional Lagrangian particle tracking routine to predict particle trajectories. Simulation results are shown to be good agreement with experimental data, where available. Based on the LES and RANS-Lagrangian methods, the mean value of the particle displacement in a straight square duct is found to generally decrease with time due to gravity effects, with the rate of deposition increasing with particle size. Using the RANS-Lagrangian method to study flows in a duct bend, there is good agreement between predicted profiles and data, with the method able to simulate particle dispersion, the phenomenon of particle roping and the increase of particle collisions with the bend-wall with particle size. With the LES-Lagrangian method, particle re-suspension from a bed is studied in a straight square duct flow and this process shown to be dominated by secondary flows within the duct, with smaller particles tending to re-suspend in preference to larger ones. Overall, the study demonstrates that modelling techniques can be used to provide insight in to processes that are of relevance to the processing of nuclear waste, and are capable of predicting their transport behaviour. In particular, they are able to provide reliable predictions of particle deposition within flows to form solid beds, the re-suspension of particles from a bed, and the influence of complex flow geometries on particle dispersion. In the latter case, they are also of value to studies of erosion due to particle impact. Such models are therefore of value as engineering tools for use in the prediction of waste behaviour and in cost effective process design.