Mathematical modelling of an aluminium spray process
Spray-forming is a newly developed industrial metal forming process in which a cylindrical metal billet is produced by the incremental deposition and solidification of an atomised metal spray on a moving substrate. A mathematical model is developed to describe billet growth and heat flow within spray-formed aluminium alloy billets. In the first part of the thesis, growth dynamics of the billet are considered. Conservation of mass at the billet surface yields a single first order quasi-linear partial differential equation for the movement of the billet surface; the nonlinearity arising from the possibility of surface shadowing. The existence of two distinctly different timescales, amongst the process motions governing billet growth, prompts the use of an averaging method. The resulting averaged equations permit analysis and are shown to provide a valid asymptotic approximation to the billet surface motion on the timescale 1/∊, for a suitably defined class of billet surfaces. The parameter ∊ ≪ 1 is the ratio of the two process timescales. Conditions under which the crown profile of the cylindrical billet becomes steady are analysed, through the averaged equations, and the stability of such profiles is examined. Computed examples of single and multiple steady state crown profiles are given. The averaged equations are also solved numerically to provide a model for transient billet growth on a "slow" timescale; results are presented. The second part of the thesis considers heat flow within the growing billet. Phase change is incorporated using an enthalpy formulation of the energy equation. The resulting equation is a nonlinear heat equation that must be solved in an expanding domain, the boundary of which is determined by solution of the billet growth model equations. Conduction on the billet length-scale takes place only on the slow timescale, with more rapid heat flow taking place only close to the billet surface. Accordingly, billet heat flow is analysed through the assumption that there is a thermal boundary layer close to the billet surface, which is driven by the "rapid" timescale spray deposition, with heat flow in the remainder of the billet driven by the time-averaged growth. The boundary layer equation is a one dimensional nonlinear advection-diffusion equation, with a nonlinear boundary condition that incorporates the intermittent deposition from the spray in the form of an irregular pulse. This equation is solved numerically using an implicit finite difference method. The slow-time heat flow is two dimensional, (assuming axisymmetric slow-time billet growth), and must also be solved numerically. For this an implicit predictor-corrector method is used. The predictor stage uses a "splitting" method, adapted from the fully implicit L.O.D. method to take account of the expanding domain. The method appears to be stable and consistent. Various numerical results are presented. The model provides significant new understanding of the dynamics of billet growth and succeeds in providing a useful framework within which the transient heat flow that occurs during spray deposition, on a number of different timescales and length-scales, can be understood. Comparison of computed model predictions with real sprayed billets confirms the validity of the model. The thesis is concluded with a summary of results and a look at possible future directions for research in this area.