Continuous precipitation of actinides is conducted in a vortex reactor at the industrial scale. The internal flow pattern produces two macro mixing zones and the concentration of the reagents strongly depends on the spatial position in the vessel. The particle size distribution is not uniform spatially either. The modelling approach developed in this work lies on previous works focussed on both the reactor hydrodynamics and the crystallization mechanisms. Previous numerical works focused on describing the reactor hydrodynamics thanks to computational fluid dynamics studies: the flow is then schematically represented as a combination of several interconnected well-mixed zones. Furthermore, crystallization mechanisms were studied to determine supersaturation expression and nucleation, growth and agglomeration kinetics.
The aim of this research work is to combine precipitation and hydrodynamics models in order to describe the behaviour of the entire vortex reactor: flow pattern (compartments), thermodynamics and crystallization kinetics are combined in a single model. The challenge consists in combining nucleation, growth and a non-conventional agglomeration mechanism of formation of loose agglomerates, with a compartmental model describing the vortex flow. A numerical framework is then developed in order to solve the steady state population balance equation and predict the characteristics of the suspension leaving the continuous vortex reactor. It lies on the combination of a minimization algorithm and an accelerated fixed point-based algorithm to predict nanoparticle and agglomerate size distributions respectively.
First, a numerical methodology to solve a single compartment is developed. This is the minimal homogeneous volume in the simulated apparatus. The steady state conditions are approached directly by the solution of the steady state population and mass balances rather than solving the corresponding unsteady equations. Furthermore, the solution methodology includes fixed point acceleration algorithms in order to determine in a reasonable computational time the crystal size distribution of the loose agglomerates. Two types of precipitation systems are tested: one including a size independent agglomeration kernel (neodymium oxalate precipitation) while in the second one, the agglomeration kernel is size dependent (uranium oxalate precipitation). The operating conditions (temperature, initial concentration, residence time, etc.) are also modified in order to guarantee algorithm robustness. In every single case, the algorithm demonstrates its performance: the mean crystal size is predicted within a tolerance of 10 μm and the error associated to the mass balances is insignificant (relative tolerance <10-4).
In order to describe the interactions between several homogeneous volumes, the developed numerical tool is first applied to the structure describing two reactors with a recycle stream. Hence a second fixed point acceleration method is included in order to achieve recycle stream convergence. In a second time, the methodology is applied to the neodymium precipitation in the 5 compartments-based model describing the vortex reactor, by considering only nucleation and growth mechanisms. Once again, the algorithm demonstrates its robustness and performance to predict the crystal size distribution in complex configurations. Indeed, the developed methodology allows for the first time to determine local supersaturation, precipitation kinetics, recovery rate and the entire crystal size distribution in each compartment of the vortex reactor.
Further works will include the application of the algorithm to describe other precipitation systems and/or multi-compartment models and the incorporation of the agglomeration process in the multi-compartment model.