Metallurgical and Materials Transactions B

, Volume 26, Issue 1, pp 159–171

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

• F. A. Acosta G.
• A. H. Castillejos E.
• J. M. Almanza R.
• A. Flores V.
Mathematical modelling

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## List of Symbols

ax

laminar coefficient (Eq. [1])

a2

turbulent coefficient (Eq. [1])

A

cross-sectional area of a pore at a givenx-position, total cross-sectional area of the filter

A, B

Cartesian axes with origin located at the pore center

Aw

window cross-sectional area

C1

ratio of cell-to-window diameters

dc

randomly distributed cell diameter (mm)

dc,max

maximum limit for the random diameter of a cell (mm)

dc,min

minimum limit for the random diameter of a cell (mm)

dw

randomly distributed window diameter (mm)

dw3

third momentum of the window diameter distribution (mm3, Eq. [28])

dw5

fifth momentum of the window diameter distribution (mm5, Eq. [25])

E(dw, θ, ϕ)

joint probability density function of the random variablesd w, θ, and ϕ

fdw

fractional frequency of windows having a diameterd w

g

gravity acceleration vector

F(C1)

geometric function defined by Eq. [29]

G(C1)

geometric function defined by Eq. [32]

H(α)

coordination number function (Eq. [33])

K

specific permeability (1 K Darcy = 10−5 cm2)

L

side length of a cubic unit lattice

mi

mass concentration of inclusions in the metal flow at the filter inlet

m0

mass concentration of inclusions at the filter outlet

mx

momentum of the fluid, in a pore, in the direction of the main flow (Eq. [13])

Mx

momentum of the fluid, in the whole filter, in the direction of the main flow (Eq. [10])

N

coordination number of a pore

Np

number of pores per unit volume of filter (Eq. [26])

NRe

Reynolds number

p

local pressure

P1,P2

pressures at the inlet and at the outlet of the filter, respectively

°p

°P

magnitude of the macroscopic pressure gradient (KPa/m) defined as the ratio (P 1, −P 2)/δ

°P1*

dimensionless pressure drop in a unit cell, at Reynolds number equal to one (Eq. [20])

°PUC*

pressure drop through a unit cell, Eq. [7]

°Puc*

dimensionless pressure drop through a unit cell, Eq. [19]

quc

voluminic flow rate, in a cell, in the direction of the main flow, Eq. [15]

Q

voluminic flow rate through the whole filter, cm3/s

Quc

voluminic flow rate through a cell, Eq. [18]

rc

radius of a pore

rs

radial distance from the axis of a pore to its wall

Rvk

local value of the v-variable (pressure or velocity) in the k-iteration

u

local fluid velocity

us

fluid superficial velocity

ux

fluid velocity component in the direction of the pore axis

[ux]x

projection of ux in the direction of the main flow

ux

mean fluid velocity, within the filter, in the direction of the main flow

uw

mean velocity of the fluid at a window

Vm

total volume of the filter

Vuc

volume of a unit cell (Eq. [A6])

Vuc

mean volume of the unit cells (Eq. [27])

Vs

volume of the spheres contained effectively in a unit lattice

Vt

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

x0

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

εc

filter critical porosity

εef

filter effective porosity

ϕ

randomly distributed azimuthal angle

μ

fluid dynamic viscosity(Kg/μm s)

v

fluid kinematic viscosity (jum2/s)

θ

randomly distributed angle formed between the pore axis and the main flow direction

ρ

fluid density (Kg/μm3)

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© The Minerals, Metals & Material Society 1995

## Authors and Affiliations

• F. A. Acosta G.
• 1
• A. H. Castillejos E.
• 1
• J. M. Almanza R.
• 1
• A. Flores V.
• 1
1. 1.Investigation Center and Advanced Studies of the IPNCINVESTAV-Unidad SaltilloCoah, Mexico