# Analysis of liquid flow through ceramic porous media used for molten metal filtration

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## Abstract

A two-dimensional mathematical model has been developed to study fluid flow inside ceramic foam filters, used for molten metal filtration, as a function of their structural characteristics. The model is based on the selection of a unit cell, geometric model, formed by two interconnected half-pores. The good agreement between experimental and computed permeabilities showed that the unit cell model approximates very well the effect of filter structure on the flow conditions inside the filter. The validity of the model is supported by the fact that permeabilities are calculated from directly measured structural parameters,*i.e.*, without the introduction of any fitting variable, such as tortuosity. The laminar flow solutions for the Navier-Stokes equation, in steady state, were obtained numerically using the control-volume method. The boundary of the unit cell was represented through axisymmetrical, body-fitted coordinates to obtain a better representation of the complex pore shape. The generality of the model, to study fluid flow in reticulated media, was tested by comparing the computed specific permeabilities with values measured for ceramic foam filters and for the new ceramic filter of lost packed bed (CEFILPB). Such a comparison shows good agreement and discloses a fundamental property of the last kind of porous medium: the critical porosity. The model indicates how porosity and pore dimensions of reticulated filters may be tailored to meet specific fluid flow requirements.

## Keywords

Material Transaction Unit Cell Model Main Flow Direction Critical Porosity Foam Filter## List of Symbols

*a*_{x}laminar coefficient (Eq. [1])

*a*_{2}turbulent coefficient (Eq. [1])

*A*cross-sectional area of a pore at a given

*x*-position, total cross-sectional area of the filter*A, B*Cartesian axes with origin located at the pore center

*A*_{w}window cross-sectional area

*C*_{1}ratio of cell-to-window diameters

*d*_{c}randomly distributed cell diameter (mm)

*d*_{c,max}maximum limit for the random diameter of a cell (mm)

*d*_{c,min}minimum limit for the random diameter of a cell (mm)

*d*_{w}randomly distributed window diameter (mm)

*d*_{w}^{3}third momentum of the window diameter distribution (mm

_{3}, Eq. [28])*d*_{w}^{5}fifth momentum of the window diameter distribution (mm

_{5}, Eq. [25])*E*(*d*_{w}, θ, ϕ)joint probability density function of the random variables

*d*_{w}, θ, and ϕ*f*_{d}_{w}fractional frequency of windows having a diameter

*d*_{w}- g
gravity acceleration vector

*F*(*C*_{1})geometric function defined by Eq. [29]

*G*(*C*_{1})geometric function defined by Eq. [32]

*H*(α)coordination number function (Eq. [33])

*K*specific permeability (1 K Darcy = 10

^{−5}cm^{2})*L*side length of a cubic unit lattice

*m*_{i}mass concentration of inclusions in the metal flow at the filter inlet

*m*_{0}mass concentration of inclusions at the filter outlet

*m*_{x}momentum of the fluid, in a pore, in the direction of the main flow (Eq. [13])

*M*_{x}momentum of the fluid, in the whole filter, in the direction of the main flow (Eq. [10])

*N*coordination number of a pore

*N*_{p}number of pores per unit volume of filter (Eq. [26])

*N*_{Re}Reynolds number

*p*local pressure

*P*_{1},*P*_{2}pressures at the inlet and at the outlet of the filter, respectively

- °
_{p} local pressure gradient

- °P
magnitude of the macroscopic pressure gradient (KPa/m) defined as the ratio (

*P*_{1}, −*P*_{2})/δ- °P
_{1}^{*} dimensionless pressure drop in a unit cell, at Reynolds number equal to one (Eq. [20])

- °P
_{UC}^{*} pressure drop through a unit cell, Eq. [7]

- °P
_{uc}^{*} dimensionless pressure drop through a unit cell, Eq. [19]

*q*_{uc}voluminic flow rate, in a cell, in the direction of the main flow, Eq. [15]

*Q*voluminic flow rate through the whole filter, cm

^{3}/s*Q*_{uc}voluminic flow rate through a cell, Eq. [18]

*r*_{c}radius of a pore

*r*_{s}radial distance from the axis of a pore to its wall

*R*_{v}^{k}local value of the v-variable (pressure or velocity) in the k-iteration

- u
local fluid velocity

*u*_{s}fluid superficial velocity

*u*_{x}fluid velocity component in the direction of the pore axis

- [
*u*_{x}]_{x} projection of u

_{x}in the direction of the main flow*u*_{x}mean fluid velocity, within the filter, in the direction of the main flow

*u*_{w}mean velocity of the fluid at a window

*V*_{m}total volume of the filter

*V*_{uc}volume of a unit cell (Eq. [A6])

*V*_{uc}mean volume of the unit cells (Eq. [27])

*V*_{s}volume of the spheres contained effectively in a unit lattice

*V*_{t}total volume of a unit lattice

*x, y*body-fitted coordinate axes located in a pore

*x*axis of a pore; position along such an axis

*X*axis in the direction of the main flow

*x*_{0}distance from the center of a pore to the center of its window

- α
limit value for θ under which axisymmetrical flow occurs

*δ*filter thickness

- °
Nabla operator

- ε
filter porosity

- ε
_{c} filter critical porosity

- ε
_{ef} filter effective porosity

*ϕ*randomly distributed azimuthal angle

*μ*fluid dynamic viscosity

*(Kg/μm*s)*v*fluid kinematic viscosity (jum

^{2}/s)*θ*randomly distributed angle formed between the pore axis and the main flow direction

*ρ*fluid density (Kg/μm

^{3})

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