# Numerical solution to the solidification of aluminium in the presence of various fibres

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## Abstract

In an attempt to understand the experimentally observed solidification microstructures in metal matrix composites, the influence of SiC, graphite and alumina fibres on the solidification of aluminium has been studied numerically. Irregular geometries of the composite material were mapped into simple rectangles through numerical conformal mapping techniques to analyse the influence of a single fibre or a row of fibres on a unidirectionally advancing planar solid-liquid interface. The fibres were assumed to be circular in cross-section and the direction of the interface movement was perpendicular to the length of the fibres. The study showed that for fibres with lower thermal conductivity than aluminium, the interface first goes through acceleration as it approaches and ascends the fibre and then deceleration as it descends the fibre. The acceleration and deceleration phenomena of the interface increases as the thermal conductivity ratio of fibre to liquid aluminium decreases. With low thermal conductivity ratios (*K*_{f}/*K*_{L}≪1), the interface is orthogonal to the fibre surface. When the conductivity of the fibre is lower than that of the melt, the interface becomes convex facing the fibre; this mode would lead to pushing of the fibre ahead if it was free to move, as has been experimentally observed in cast microstructures of metal matrix composites. The temperature versus solidification time plots of two points, one in the fibre and the other in aluminium, show that the fibre with a conductivity lower than the matrix is at a temperature higher than the melt; the temperature difference between the two points increases with increasing solidification rate for all the positions of the interface before it touches the fibre. The three-fibre study shows that as the number of fibres increases, the curvature of the interface increases upon approaching the subsequent fibres. The relationship between these numerical computations and experimental observations has been discussed.

## Keywords

Conformal Mapping Metal Matrix Composite Liquid Aluminium Interface Movement Solidification Microstructure## Nomenclature

*a*reference length = diameter of the fibre

*h*\(\begin{gathered} Jacobian = \left( {\frac{{\delta x}}{{\delta \xi }}} \right)^2 + \left( {\frac{{\delta y}}{{\delta \xi }}} \right)^2 \hfill \\ = \left( {\frac{{\delta x}}{{\delta \eta }}} \right)^2 + \left( {\frac{{\delta y}}{{\delta \eta }}} \right)^2 \hfill \\ \end{gathered} \)

*K*thermal conductivity; in Equation 4 it is defined as

*K*″ = (*K*+*K*_{f})/2 for the common boundary between fibre and the freezing medium. For all the rest of the points*K*″ =*K*in Equation 4*L*latent heat of fusion

*r*non-dimensional variable in radial direction

*S*non-dimensional distance travelled by the interface

- Ste
Stefan number = \(\frac{{C_{ps} \left( {\mathop T\nolimits_m^* - \mathop T\nolimits_0^* } \right)}}{L}\)

*T*non-dimensional temperature

*t*non-dimensional time

*x*a non-dimensional spatial coordinate of physical plane

*y*a non-dimensional spatial coordinate of physical plane

- α
thermal diffusivity

- η
non-dimensional axial coordinate of the mapped plane

- ξ
non-dimensional vertical coordinate of the mapped plane

- θ
a polar coordinate

- l
liquid

- m
melting

- s
solid

- O
constant wall temperature

- i
initial

- f
fibre

- *
dimensional variables

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## References

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