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Abstract
The permeability of the semi-solid state is important for the compensation of volume shrinkage during solidification, since insufficient melt feeding can cause casting defects such as hot cracks or pores. Direct measurement of permeability during the dynamical evolution of solidification structures is almost impossible, and numerical simulations are the best way to obtain quantitative values. Equiaxed solidification of the Al-Si-Mg alloy A356 was simulated on the microscopic scale using the phase field method. Simulated 3D solidification structures for different stages along the solidification path were digitally processed and scaled up to generate 3D models by additive manufacturing via fused filament fabrication (FFF). The Darcy permeability of these models was determined by measuring the flow rate and the pressure drop using glycerol as a model fluid. The main focus of this work is a comparison of the measured permeability to results from computational fluid flow simulations in the phase field framework. In particular, the effect of the geometrical constraint due to isolated domain walls in a unit cell with a periodic microstructure is discussed. A novel method to minimize this effect is presented. For permeability values varying by more than two orders of magnitude, the largest deviation between measured and simulated permeabilities is less than a factor of two.
Highlights
IntroductionThe formation of defects such as shrinkage porosity [1] or hot tears [2,3] depend on melt flow through the mushy zone, and on its permeability.Due to the small scale of length of the melt-filled structures, the Reynolds number is typically very low
In casting processes, the formation of defects such as shrinkage porosity [1] or hot tears [2,3] depend on melt flow through the mushy zone, and on its permeability.Due to the small scale of length of the melt-filled structures, the Reynolds number is typically very low
Since the permeability scales proportional to the square of the feature size, the measured results were multiplied by a factor of α−2, to represent the permeability of the original microstructures
Summary
The formation of defects such as shrinkage porosity [1] or hot tears [2,3] depend on melt flow through the mushy zone, and on its permeability.Due to the small scale of length of the melt-filled structures, the Reynolds number is typically very low. The Reynolds number describes the ratio of the inertial forces over the viscous forces in a flow, for a very low Reynolds number the contribution of the inertial term in the Navier–Stokes equations can be considered negligible Omitting this term leads to the Stokes equations, which, for an incompressible fluid and a steady state solution, take the form: μ∇2u − ∇p + f = 0. Since solutions (u, p) of the associated homogeneous system can be superimposed linearly, this implies a linear correlation between the averaged pressure gradient ∇p and the averaged flow velocity u (the superficial velocity) for a region. This linear correlation is described by Darcy’s law:
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