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# 2 µm - eTheses Repository - University of Birmingham

2 µm - eTheses Repository - University of Birmingham

## where γHg is the

where γHg is the surface tension of mercury, θHg is the wetting angle of mercury on ceramics and P is the applied pressure. It describes the dynamic equilibrium between external forces tending to force a liquid into a capillary of diameter DHg and the internal forces repelling entry into the capillary. According to Rootare and Prenzlow (109) the surface area from mercury porosimetry SsHg can be calculated from: S sHg = γ Hg 1 cosθ Hg Vtot ∫ 0 PdV 41 Equation 22 where Vtot is the total intrusion volume, P is the pressure and V is the volume of the incremental intrusions. Based on porosimetry data, Leon (110) proposed a simple relation between the product of mean volume pore diameter DHg and the permeability KMIP of a porous body represented by Equation 23 with φ the powder bed porosity. 2 φ DHg K MIP = Equation 23 32 In contrast to the static intrusion when using mercury intrusion porosimetry, the preform infiltration is done in a dynamic way. Thus different approaches for modelling of the fluid flow in infiltration were considered. 2.3.2. Fluid flow in preform infiltration Fluid flow in a porous medium depends on the properties of the medium to be penetrated and the fluid properties. A simple model for infiltration of porous media is given by the Darcy equation. Henri Darcy established empirically that the flux of water through a permeable formation is proportional to the distance between the top and bottom of the porous column.

The superficial velocity v0 in the flow direction z is calculated in respect to the pressure gradient dP/dz at the infiltration front and the fluid viscosity µ: Ks dP = − ⋅ μ dz v Equation 24 0 The constant of proportionality is called the specific intrinsic permeability Ks (111) . Darcy's equation represents a simplification of the general equation of viscous fluid flow governed by the Navier-Stokes equation. The simplification is made by assuming incompressible fluids, laminar flow, and unidirectional, saturated flow. The superficial Darcy velocity v0 is a macroscopic concept, and is easily measured. It should be noted that Darcy’s velocity is different from the microscopic velocities associated with the actual paths of individual volume elements of molten metal as they wind their way through the pores in the preform (103) . Darcy´s equation is limited to saturated flow. Thus complete saturation before further through-penetration is assumed. Particulate preforms used to produce MMCs generally have highly complex internal void geometries. This complexity and capillarity during infiltration render prediction of the metal flow path during infiltration too complicated to be realistically modelled at the microscopic level of the individual particles that make up the preform. During preform infiltration, a number of physical, mechanical and chemical phenomena interact, including multiphase flow of liquid metal and air in a porous medium, heat and mass transfer related to solidification, equilibrium of mechanical forces, and chemical interfacial reactions between reinforcement and matrix. Analytical and numerical solutions have been given and compared to experimental data for (i) unidirectional infiltration under constant applied pressure, including non-isothermal infiltration by a pure metal or a binary alloy (112) , 42

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