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., and A. FLORES V. 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. I. INTRODUCTION CERAMIC foam filters are playing an increasingly important role in the physical purification of metals, both in the ferrous and nonferrous industries. Certainly, the latter industry is the largest consumer of this filtering media, particularly the aluminum casting industry. Aubrey and Dore o~ reported that in 1992, eight million metric tons, equivalent to - 5 0 pct of all the aluminum produced in the world, was filtered using ceramic foam filters. The use of foam filters is also rapidly spreading toward filtration of high-temperature alloys. Sutton et al.12I have reported results on the filtration of superalloys, and several articles present issues of plant and laboratory tests in the filtration of cast irons and steels. [3-6] Recently, Gating and Cummingst6~announced the successful filtration of carbon and alloy steels using ceramic foam filters, in the tundish of a four-strand bloom caster. The results from these works look so promising that one could expect that filtration may soon become a clean steel-making practice. The wide acceptance of ceramic foam filters is based on their structural properties, which allow the filter, among other things, to have a low flow resistance and a high filtration efficiency. The webs (ceramic elements of the structure) of these filters create a tortuous path for the fluid to flow through favoring the probability of inclusion contact with the internal surfaces of the filter; i.e., for inclusions much smaller than the pore size, the foam filters provide deep bed filtration. The filtration efficiency of foam filters is closely related to the fluid flow conditions through the porous medium, tT] Fluid F.A. ACOSTA G. and A. FLORES V., Assistant Professors, A.H. CASTILLEJOS E., Associate Professor, and J.M. ALMANZA R,, Research Assistant, are with the Investigation Center and Advanced Studies of the 1PN, CINVESTAV-Unidad Saltillo, Saltillo 25000, Coah, Mexico. Manuscript submitted November 9, 1993. METALLURGICAL AND MATERIALS TRANSACTIONS B flow plays an important role in transporting inclusions to the filter wall where they can attach, but also, it can be responsible for dragging out the captured inclusions. Several publications indicate that filtration efficiency decreases with increasing superficial melt velocity, t1'7'8] Aubrey and Dore tl] found that foam filters, having a pore size of 30 ppi (pores per inch) and a thickness of 2 inches, worked efficiently in the filtration of aluminum at mass fluxes between 70 and 200 g/cm 2 rain, i.e., for aluminum melt superficial velocities between 0.5 and 1.5 cm/s. They observed that capture efficiency decreased dramatically above that range. From the previous arguments, it is clear that it is desirable to be able to predict the flow rate obtainable under a given pressure drop or to be able to predict the pressure drop necessary to achieve a specific flow rate. In fluid mechanics of granular porous media, the relationship between these quantities is represented by the Ergun equation, IVP] = ( a l + a2Q)Q [ll where al ( = I x / K A ) and a2 are coefficients that account for laminar and turbulent effects, respectively. The Darcy viscous specific permeability, K, can be determined either experimentally by measuring pressure gradient vs flow rate or theoretically from packing structure properties. However, for ceramic foam structures, there is not a reported relationship between structure properties and specific permeability. Under these circumstances, ceramic foam filters manufacturers have generated experimental data showing indirectly, the permeability of their products as plots of percent of original flow rate vs filter area/choke area ratio, for filters of different pore sizes, t3] The metal flow rate is held at a value close to the no-filter situation by using filters with an area several times larger than the choke area of the gating system. Proper filter sizing is also important to VOLUME 26B, FEBRUARY 1995--159 avoid premature filter blockage by inclusions. [3'6] Sutton et al.tZ] have indicated that the resistance of a given filter to metal flow can be reduced considerably by increasing the pore size and porosity and by decreasing the web thickness, but this is achieved at the expense of weakening the filter structure. The authors indicate that to explain differences in the permeability of filters is necessary to consider the complex effects that pore shape and size, porosity, and webs structure have on it. Filter permeability also plays a role in promoting flow regulation downstream the filter, i.e., the achievement of laminar flow conditions under specific flow rates. 19] Flow regulation acts in preventing the penetration of vorticity in the inlet flow, in reducing turbulent entrainment of foreign materials, and in limiting metal reoxidation and mold erosion. The complexity of most porous media makes impossible the exact description of the solid boundaries that are determinant for the characteristics of fluid flow inside the filters. Therefore, before pursuing a microscopic model to determine filtration efficiency, it is necessary to obtain a geometric model that resembles the pore structure and also approximates its overall effect to flow. Being that the specific permeability is an intrinsic property of the structure of porous media, it has been used by Payatakes and co-workers l~o,~~,12]to validate geometric models of granular porous media and fluid flow behavior within them. Payatakes et a/. tl~ developed a model for isotropic monosized granular porous media based on a single unit cell. The model considered the convergent-divergent character of the flow ducts and the random dimensions of pores and constrictions and was used t~] to carry out trajectory calculations of particles in deep bed filtration. In a subsequent work, l~2]the unit cell model was further improved by considering the random orientation of pores and constrictions. The authors calculated the flow field within the unit cell and from it the permeability, and they found an excellent agreement between experimental and calculated permeability values. In a different approach to the unit cell model, Tian and Guthrie t~3] studied fluid flow in porous media by representing the ceramic web structure of foam filters by a number of orderly placed cylindrical obstacles. The porosity of the porous media was prescribed by varying the number density of obstacles. This geometric model was used to calculate the fluid flow field within the filter and from it the permeability. Apparently, the fitting of predicted with measured permeabilities was used as the criterion for the placement and sizing of the obstacles representing the filter walls. Gauckler et al. 1t4~ used the method of Payatakes et al. ~ ] to investigate the filtration behavior of ceramic foam filters to purify molten aluminum. However, the investigators did not give details on the flow patterns used in their filtration efficiency calculations. Engh et al. I~5I developed a theory to study the deposition of solid particles on single spherical or nonspherical collectors and extended it to determine collection efficiencies for packed beds of spheres. The effect of the roughness of the collector surface on collision efficiency has been discussed in the literature. 1~6,~71 The results reported in this study represent the start of a long-range investigation whose objective is the development of a comprehensive theory of deep bed liquid 160--VOLUME 26B, FEBRUARY 1995 filtration with foam filters. Such a theory requires a geometric model that resembles the pore structure of the filter media and also approximates its overall effect to flow. This article addresses the problem of creeping Newtonian fluid motion in ceramic foam filters and in ceramic filter of lost packed bed (CEFILPB) t8"~8'191 or pore-formed ceramic filters, t2~ The approach is based on the selection of a unit cell formed by two interconnected half-pores. The unit cell is considered a section of a periodically constricted tube with average dimensions and orientation. These geometric parameters are obtained from the measured distributions of pore and window (interconnecting openings between pores) diameters and by considering the random orientation of the pores. For accurate modeling of the fluid flow within the complex geometry of the unit cell, a bodyfitted coordinate system was employed. The governing Navier-Stokes equation was solved numerically using a control-volume technique. This work generalizes the ideas of Payatakes et al. by introducing a relationship between the porosity, the average pore coordination number (number of windows in a pore or number of closest neighboring pores to a given one), and the window-to-pore diameter ratio, all of which have an effect on the motion of fluid through porous media. To validate the proposed model, the permeabilities of foam and CEFILPB filters were measured and compared with the corresponding calculated values. A good agreement was found, and the model helped to disclose the existence of a critical porosity in pore former filters. The modeling of the pore space and of the fluid flow within it forms the basis for the study of deep bed filtration, which is presently pursued. II. FORMULATION OF TIlE PROBLEM A. System Considered The fluid flow and permeabilities of the ceramic porous bodies shown in Figures l(a) and (b) have been investigated. These are filters of high porosity (>70 pct). The fabrication process for the foam filters involves coating a polyurethane foam with a ceramic slurry that is let to dry for later burn off of the polymer. The process results in a positive replica of the foam where the pores are surrounded by interconnected empty strands of ceramic. On the other hand, the CEFILPB, or pore former filters, are obtained by saturating the interstitial spaces of a packed bed of particles with a ceramic slurry. After the slurry has dried out and the ceramic structure has achieved enough green strength, the particles are leached out. This is the reason for the name lost packed bed filters. The process leads to a ceramic structure that corresponds to the negative of the packed bed structure. The nature and flexibility of this process allow it (a) to obtain compact ceramic walls; (b) to control porosity, pore size, and pore shape; (c) to avoid flow channeling; and (d) to have small interconnecting windows between the pores. The control on the dimensions of the flow passages is very useful for tailoring permeabilities. METALLURGICAL AND MATERIALS TRANSACTIONS B (a) (b) Fig. 1 - - P h o t o g r a p h s of views perpendicular and parallel to the direction of flow of (a) CEFILPB and (b) foam filters. B. Experimental Work The experimental work consisted of measuring filter permeabilities to validate the calculations and determining pore and window sizes and porosity, which are the parameters needed by the mathematical model. The specific permeabilities were measured in filters of a 50-mm diameter and 20-mm thick, using water flowing in a closed loop, as that shown in Figure 2. Pressure drops for different water flow rates were measured twice, once for each filter face oriented upstream. Figure 3 shows plots of macroscopic pressure gradient vs water flow rate obtained for a CEFILPB and a foam filter. From this figure, it is seen that the measured zoo 1 r l *~ eO CEFILPB--4+2.7~C=0.86 :Z 240 9 Foam .o } l I i 40 ppi, E = 0 . 8 6 200 Q) 9 160 Flowmeter lzo 80 Pump ~ Filter WaterReservoir U Manometer Fig. 2--Scheme of the closed water loop used for measuring permeability of filters. METALLURGICAL AND MATERIALS TRANSACTIONS B ~O 40 :~ 0 i 0 100 200 300 400 500 Water flow rate Q, e m a / s 600 Fig. 3--Plots of measured pressure gradient as a function of water flow rate. Two flow directions were tested for every filter as indicated by the empty and full symbols. VOLUME 26B, FEBRUARY 1995--161 pressure gradients are essentially independent of the surface facing upstream, indicating that the filters are isotropic. Figure 3 also indicates that, as expected from Ergun's equation for porous media (Eq. (1)), the pressure gradient through the filter follows a parabolic relationship with the flow rate. The specific permeability, K, was determined from the experimental results by fitting a parabola to them and computing the slope of the ]re[ vs Q curve at zero flow rate, i.e., laminar flow conditions. Table I presents the mean specific permeabilities measured on particular foam and CEFILPB filters of different nominal pore sizes. For ceramic foam filters, the nominal pore size is given in pores per inch (ppi), while for CEFILPB filters it is given in terms of the diameter of the pore former. Table II reports additional foam filter permeabilities measured in this work and in those of Tian and Guthrie [13j and Sane et al. [2~ From the table, it is seen that permeability may vary appreciably for filters of the same nominal pore size and that porosity alone does not explain the observed changes in permeability values. As shown later in this section, the cell and window size distributions also play a role in determining the permeability of the porous medium, i.e., fluid flow behavior. The pore and window size distributions were obtained optically with the help of an image analysis system using an intercept length method. For the determination, the filters were impregnated with a transparent resin before cutting them along a plane parallel to the direction of flow; the resin avoided breakage of the ceramic walls. The Spektor's method was utilized to determine pore diameter distribution from measured chord lengths. Iz3] Since the pores are slightly ellipsoidal, the chords were measured, both, in the parallel and perpendicular directions to the flow, and the mean pore size was defined as the average of the two mean diameters. The average Table I. Nominal Pore Size 30 ppi 45 ppi 40 ppi 50 ppi C. Mathematical Formulation 1. Representation of the filter structure From the photographs appearing in Figure 1, it is clear that the complexity of the porous structure would make it quite difficult to specify the location of the pore walls and, therefore, of the boundary domain. Furthermore, if the entire filter was used as the calculation domain, the required computer storage and computer time would be truly excessive. A useful simplification arises by assuming that the fluid flow is fully developed, such that the velocity field repeats itself pore after pore. However, Experimentally Determined Properties of Ceramic Foam and CEFILPB Filters Nominal Pore Size 10 ppi 15 ppi 20 ppi 30 ppi 40 ppi 50 ppi -2.7 + 2 mm - 4 + 2.7 mm -6.3 + 4 mm Table II. pore and window sizes for particular filters of different nominal size are given in Table I. The window diameters were determined by measuring the diameters of complete windows located parallel to the cut plane and just below it, examples of this type of window are indicated by the arrows appearing in Figure 1. A typical window size distribution for a foam filter is shown in Figure 4. The window size distributions allowed one to obtain the statistical quantities required by the model, the third and fifth momenta of the window size. The pore porosity, e, was measured according to the specification ASTM C20-87 applicable to ceramic refractories. In this method, the porosity is obtained by dividing the weight of water retained in the macropores of the filter by the weight of water that occupies a volume equivalent to that of the whole filter. The porosity values for randomly chosen samples of CEFILPB and foam filters are presented in Table I. Figure 5 shows these porosity values plotted as a function of (dw)/(dc). The black dots, corresponding to pore former filters, show that porosity increases with the increase in diameter ratio; this relationship is somewhat irregular in the case of foam filters. K (KDarcy) 43.0 45.3 38.9 16.0 14.4 7.8 4.8 6.0 19.3 (d~) (mm) 3.85 3.15 2.55 2.06 1.37 0.98 1.75 1.93 3.41 (dw)/(dc) 0.44 0.36 0.45 0.45 0.5 0.54 0.4 0.43 0.57 (d~) (mm) 1.69 1.13 1.15 0.93 0.69 0.53 0.71 0.83 1.94 e 0.8 0.89 0.82 0.88 0.86 0.88 0.79 0.8 0.85 Typical Porosities and Specific Permeabilities of Ceramic Foam Filters, Reported by Different Authors e 1161 0.78 0.87 0.81 0.88 --- 162--VOLUME26B, FEBRUARY1995 K (KDarcy)1161 e (Present Work) K (KDarcy) (Present Work) 6.96 15.02 3.75 9.59 --- 0.87 0.88 -. 0.86 0.88 14.3 16.0 -. 14.4 7.8 . eI l 2 j K (KDarcy)l~2J 0.85 0.89 0.8 18 40 10 --- --- . METALLURGICALAND MATERIALSTRANSACTIONSB 1.0 I I I I I I I >~176 / 0.9 0.8 ,.o E= 4 9 - 0.7 Z tu 0.6 0.5 0 o 0.4 0.3 -- 11 0.2 3 Computed 9 Measured in C E F I L P B O Measured in f o a m filters 0.1 0.5 1.0 1.5 0.0 0.0 Window Diameter, & ( m m ) Fig. 4 - - H i s t o g r a m showing typical window diameter distribution of a foam filter. this task needs the adequate selection of a geometrical pore module that reflects the pore coordination number, the random distribution of pore and window diameters, and orientation and curvature of the pore wails. The coordination number can be estimated from visual inspection of sectioned filters. However, a more reliable method can be obtained by considering different packing modes of the pore former and recognizing that the packing fraction, i.e., porosity, is related to the ratio dw/dc and to the coordination number, N. As mentioned in Section A, CEFILPB filters are made by impregnating a packed bed of spheres with a ceramic slurry. When these particles are leached out, the remaining ceramic body has a porosity equal to the particle packing fraction of the original bed. From geometry, the packing fraction of the arrays of spheres shown in Figure 6, for coordination numbers 4, 6, and 8, are 0.34, 0.52, and 0.68, respectively. In these unit lattices, rigid spheres are in mutual contact in just one point; i.e., the resulting adjacent pores would have a window diameter equal to zero. However, CEFILPB filters are formed by wellinterconnected pores that are produced from spheres being in mutual contact over a finite area. To determine the dependence of the window size on the packing fraction, let us consider the overlapping between two neighbor pore former spheres when the distance between their centers decreases, from a distance de, the pore diameter, to a distance equal to 2x0, the distance between the centers of interconnected pores. The size of the window appearing depends on the degree of overlapping (Appendix). The relationship of the porosity with N and dw/dc can be obtained considering the following definition: Vs v, e = -- [2] where V, is the volume occupied by the spheres contained within a unit lattice of volume V, and side length METALLURGICAL AND MATERIALS TRANSACTIONS B I I I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 dw/d c Fig. 5--Computed and experimentally determined porosities as a function of window-to-ceU diameter ratio, for sphere arrangements with differentcoordination numbers. (a) (b) (c) Fig. 6 - - U n i t cell lattices showing spheres with different coordination numbers: (a) diamond cubic lattice, N = 4; (b) cubic lattice, N --- 6; (c) body-centered cubic lattice, N = 8. L (Figure 6). Table III shows the parameters used to calculate the porosity for lattice arrangements of different coordination numbers. For the calculations, it is assumed that, despite their overlapping, the volume of the spheres remain constant. Figure 5 shows the calculated filter porosity as a function of dw/dc for three coordination numbers. The figure indicates that a larger porosity, i.e., a closer packing of the pore former, results VOLUME 26B, FEBRUARY 1995--163 Table III. Lattice Parameters Used to Calculate Filter Porosities Lattice Denomination Diamond cubic Cubic Body-centered cubic Coordination Number N Number of Spheres per Lattice Cube Side Length L 4 6 8 1 4.62Xo 2x0 inlet and one outlet and which incorporates the flow constriction represented by the window. Thus, the unit cell selected to represent foam and CEFILPB filters is built by two half-pores, as it is shown in Figure 8. The unit cell has an inlet and an outlet having a diameter equal to the pore diameter and a constriction surface with a diameter equal to the window diameter. The length of the unit cell can be expressed as (Appendix) 8 2 2.3 lxo L = 2Xo = dw(C2 - 1) 1/2 in higher values of dw/d~ for a given coordination number. A comparison of the experimental values of e vs dw/d~ with the calculated curves indicates that the coordination number for both foam and CEFILPB filters is close to six. Visual inspection of the filters structure reveals a similar value. Figure 7 displays schematically a single pore having windows centered on the axes A, B, and C, which corresponds to a coordination number of six. In the figure, the X-axis indicates the main flow direction, and 0 represents the random angle between the pore axis, C, and the X-axis. Due to the complexity of the filter structures, the pores are randomly oriented; i.e., 0 can take values in the range from 0 to 7r/2. When the angle 0 has a value equal to zero, the whole flow occurs through the windows located on the C-axis. This axis remains the preferred flow direction for angles 0 -< 0 < t~. The angle a occurs when the axes A, B, and C are equally inclined with respect to the main flow direction, and the fluid flow is equally probable along any of these three axes. From geometry, it is found that for a pore coordination number of six, a -- 54.74 deg. With this consideration in mind, we can postulate that the fluid flow in the macroscopic transport direction can be represented by the flow occurring through a unit cell having only one [3] where C1 = dc/dw. It is important to mention that de, dw, and x0 are related through only one constant, C1, since the pore geometry is spherical. Nonspherical pores require more than one constant to relate their internal dimensions, as is the case in packed bed filters, tm'~t'121 2. Assumptions, governing equations, and boundary conditions For the purpose of calculations, it is assumed that the unit cell for foam and CEFILPB filters is the same despite the differences in their method of fabrication. Additionally, the following assumptions are made: (1) the fluid flow through any unit cell is fully developed; i.e., there is a common pressure drop across any unit cell within the filter; (2) the fluid flow is laminar; (3) the fluid flow is axisymmetrical; this assumption is acceptable because the fluid moves predominantly in the main flow direction as pointed out previously; and (4) the fluid flow through the unit cell occurs under steady-state conditions. The governing differential equations for the velocity and pressure fields within the unit cell are given by the continuity and Navier-Stokes equations, which are written as V.u = 0 - - ~ p -- V ( p U : U ) "t- ].tV2U -[" p g = 0 [4] [5] The boundary conditions are specified as follows: 1. Axis of the unit celt: zero flux conditions are imposed as a result of axisymmetry -- Or O=a = 0 u, = 0 2. Unit cell walls: nonslip and impermeable conditions are imposed; i.e., flow velocities tangential and normal to the wall are zero. 3. Inlet and outlet: across any given unit cell, there is a prescribed common pressure drop, APuc, defined as -APtJc = [VPl2xo cos 0 A [7] Where IVP] is the magnitude of the macroscopic pressure gradient acting on the filter and Xo and 0 are random variables, which meanings have previously been given (Section C - 1). X Fig. 7 - - Schematic of a pore with a coordination number of six, illustrating the random nature of pore orientation with respect to the main flow direction. 164--VOLUME 26B, FEBRUARY 1995 [6] 3. Derivation of the mathematical expression for the permeability In order to validate the fluid flow calculations under laminar flow conditions, an expression for the specific permeability, K, as a function of the geometry of the METALLURGICAL AND MATERIALS TRANSACTIONS B Main Flow Direction FILTER . LL AXIS UNIT ICELL WINDOW ~ F ain l o w -- / o~el Direction J axis 9 Fig. 8--Selected unit cell for CEFILPB and foam filters, showing the orientation angles O and ~b and the characteristic dimensions dw, de, and 2Xo. porous structure has to be obtained. From Darcy's law, it is known that ]Ve] us = K /x [8] and from continuity, the superficial velocity, us, is us = (ux)e [9] where (Ux) is the mean fluid velocity, within the filter, in the direction of the main flow, i.e., X-direction. This velocity can be computed from the X-momentum of the fluid as it passes the filter, Mx = oeV,.(Ux) [101 which in tum, it is defined as Mx = V,,,Np fe mx(dw' (9)dE [11] where Vm is the volume of the porous medium, Np is the number of pores per unit volume, mx is the X-momentum in a unit cell, dw is the random window diameter, and METALLURGICAL AND MATERIALS TRANSACTIONS B dE is the joint probability density function of the random variables dw, O, and ~b that characterize the unit cell, dE(dw, O, r~) = sin 0 dfbdOdd,~ - cos a) 2r [ 12] The X-fluid momentum in the unit cell is calculated from the fluid flow field according to the following expression: mx = p f [Ux]xdv [13] . I v UC where [Ux]x is the projection of the ux velocity in the X-direction. For the purpose of this work, Eq. [13] can be written as mx=pfo2XO{fA[Ux]xdA}dx [14] where A is the cross-sectional area of the pore at any given x position. The quantity inside the key bracket is the voluminic flow rate in the X-direction, quc, which is VOLUME 26B, FEBRUARY 1995--165 related to the voluminic flow rate through the cell, Quc, according to the following expression: (ClSw)= dSwfdwddw w, qvc = Quc cos 0 [15] the number of pores per unit volume is given as therefore, E Np - 2x0 mx = p Quc cos Odx [25] in (Vuc) [16] [26] where the average unit cell volume is (VtJc) = F( C1)(d3w) and since Qvc is, independent of x, then mx = p2xoQoc cos 0 [17] The term Qoc can be calculated from the velocity field obtained by solving Eqs. [4] and [5]. For convenience, the flow rate is written in terms of the mean flow velocity and cross-sectional area of the window, Uw and Aw, respectively, Quc = u,,Aw z~euc AP*c = - - pU:w ~ dw.max (d3w) 3 d;fdfldw and the function F(C~) (Appendix) is f(CO = 8 2C~(C~ - 1) 1 / 2 - ~ ( C ~ - l ) 3/2 Np - F(CI)(d3 ) [20] [31] where C~ - 1 G(C1) - - - (2Xo)21VPIcos 0 [21] H(a) = 1 + cos a + cosZa mx = /z(-~tP*) [22] Substitution of Eqs. [22] and [3] into Eq. [11] results in the following expression for the total X-momentum of the fluid within the filter: VmNppIVPI(C ~ - 1)3/27r f Mx = 4-~---~*) Je d~ cos20 dE zXP*. [23] Evaluating the integral over the limits ~b E {0, 2~r}, 0 {0, a}, and dw E {dw,min, d w. . . . }, the following expression is obtained: cos20 dE = (1 + cos a + cos 2 ~) 3 [24] where (dSw) is the fifth momentum of the window diameter distribution and is expressed as 166--VOLUME 26B, FEBRUARY 1995 [33] The importance of Eq. [31] is that the specific permeability can be calculated in terms of easily measurable parameters, i.e., the porosity, e; the third and fifth momenta of the window diameter, (d3w) and (dSw); the ratio of mean pore diameter to mean window diameter, C~; the limit angle a, which is a function of the coordination number; and the computed dimensionless pressure drop, III. s [32] + l and then, p(Zxo)3lVPlcosZOAw [30] Finally, substitution of the Eqs. [9], [10], [23], [24], and [30] into Eq. [8] results in the following expression for the specific permeability, K, K = G(CI)H(a) /z(-AP*) [29] From Eqs. [26] and [27], the number of pores per unit volume of filter can be rewritten as [191 The Reynolds number has been defined as NR~ = 2XoUwp/p. From Eqs. [7], [19], and [20] and from the Reynolds number definition, the following expression is obtained: uw = [281 19 dw,min E and (b) the dimensionless pressure drop corresponding to a Reynolds number equal to one, Z~191* = NReZkPu*c (d3w) is the third momentum of the window diameter, defined as [181 An expression for Uw in terms of the structural parameters of the unit cell can be obtained introducing the following definitions: (a) the dimensionless pressure drop across a unit cell, [271 SOLUTION PROCEDURE The differential equations of mass and momentum conservation were solved using the control volume method implemented in the PHOENICS code. An accurate representation of the unit cell surface and therefore a reliable prescription of the boundary conditions at this surface was achieved using the body-fitted coordinate system capability included in PHOENICS. Poisson's equation was solved to transform the coordinate system from the physical plane to a computational METALLURGICAL AND MATERIALS TRANSACTIONS B plane and generate a mesh with improved grid orthogonality. The method of orthogonal attraction over the wall boundary was chosen, specifying that the grid lines normal to that boundary are orthogonal to it. Two nonuniform grids, (30 • 10) and (48 • 24), were used in the computations to assess the grid size sensitivity. Only minor differences in the results obtained from both meshes were observed, and therefore, most of the calculations were done with the coarser grid shown in Figure 9. This mesh allowed time savings without sacrificing accuracy. Convergence was based on the following criterion: [34] Max(IRkv+ 1 -- R v, l / I n v k+, I) ~ 1 x 10 -4 where R~ is the local value of the v variable (pressure or velocity) in the k iteration. It was found that the convergence of the numerical solution, in the tiny flow domain under consideration, was favored by the use of microns as the length dimension. The procedure for computing the dimensionless pressure drop, APt, was to solve Eqs. [4] and [5] for different values of APuc, i.e., different boundary values at the inlet and outlet, until the flow velocity profile at the window resulted in a mean average velocity such that the Reynolds number was equal to one. IV. >: 800 m l c / s e c ~ - - R E S U L T S AND DISCUSSION Figures 10(a) and (b) show typical fluid velocity and pressure fields corresponding to an NR~ = 1, respectively. The fields are computed for water flowing inside the unit cell of a filter with a dw/dc = 0.4 and a length L = 1.6 mm; these dimensions belong to the CEFILPB filter with a nominal size of - 2 . 7 + 2 mm reported in Table I. It should be made clear that the same calculations would be valid for a foam filter with identical values of the structural parameters dw/dc and L. Although the unit cell was derived from analyzing the pore former filter structure, the permeability calculations given below demonstrate that it is also valid for representing the pore structure in foam filters. Figure 10(a) shows that for laminar flow conditions, there is not recirculatory flow; also, there is a strong variation of the Inlet velocity along the axial direction, and large quasistagnant zones occur close to the pore wall. Figure 10(b) indicates that the pressure drop, through the unit cell, happens mainly in the vicinity of the window. The dimensionless pressure drop, AP*, through a unit cell was determined considering the flow of aluminum, water, and air through the unit cell. The physical properties of the three fluids are given in Table IV. It was found that the values of dimensionless pressure drop for the three cases did not differ by more than 5 pct. This behavior indicates that AP* depends exclusively on pore structure and not on fluid properties. Thus, from Eq. [31], it is clear that also the specific permeability, K, is only a function of the pore structure. These results are in agreement with that found in the literature for granular porous media, t24] Table V reports the dimensionless pressure drop for filters of different nominal pore size. The computation of the pressure drop was also Pressure=O0156 Po Outlet Velocity at L , the ~all~O Pressure=OC / - 0 i (a) ~,/// 6 11 (b) Pa , ~2 / Fig. 1 0 - - P r e d i c t e d (a) mean velocity field and (b) bution for water flowing, at a NR, = 1, through a unit dw/d,: = 0.4 and a length L = 1.6 m m . The numbers indicate relative pressure uniformly spaced from 3 x 10 -3 Pa. pressure districell with a ratio on the contours 10 -2 to - 6 . 6 • --t--pq,~ ~ - . Table IV. Fluid Physical Properties Used in Permeability Calculations -t-4--4~ A~IS 3r 5''P~er~T7 ~ , . . . . . Fig. 9 - - S c h e m a t i c of axisymmetric unit cell system, illustrating the 30 • 10 body-fitted grid and the boundary conditions. METALLURGICAL AND MATERIALS TRANSACTIONS B Kinematic Viscosity ~, (/zmZ/s) Fluid Air Density p (Kg//zm 3) 1.200 X 10 -18 Water 9 . 9 8 2 x 10 -16 1 . 5 0 X 107 1.01 x 106 Aluminum 2.300 • 10-15 5.17 • 105 VOLUME 26B, FEBRUARY 1995--167 Table V. Parameters Involved in the Model to Compute Specific Permeability, Equation [32], H(a = 54.74 Deg) = 1.91055 Nominal Pore Size 10 ppi 15 ppi 20 ppi 30 ppi 40 ppi 50 ppi -2.7 + 2 mm - 4 + 2.7 mm -6.3 + 4 mm G(CO (dSw)l(d3) ~ 0.37 5 0.8 0.41 3 0.89 0.36 2.1 0.82 0.36 1.6 0.88 0.33 0.78 0.86 0.31 0.55 0.88 0.39 1.22 0.79 0.37 2.01 0.8 0.29 8.72 0.85 14 12 - ~ * K (KDarcy) 34.1 81.8 46 45.3 39.3 30.2 38.4 24.9 27 15.8 23.9 11.9 39.6 18.1 38.7 29.6 31.1 131.5 done for a unit cell belonging to a packed bed filter, which permeability has been reported before, tm~ The numerically and analytically computed AP* values, for a packed bed of glass spheres, did not differ by more than 10 pet. t~~ Table V presents, for different filters, the values of the parameter involved in the calculation of the specific permeability according to Eq. [31]. From the table, it can be seen that the specific permeability varies directly with the ratio of fifth-to-third momenta of the window size distribution. Although porosity, dimensionless pressure drop, and G(C]) have an effect on permeability, no simple relationships can be observed from these data. Figure 11 presents calculated values of the dimensionless specific permeability, K* = K/d~, as a function of dw/d~. The plots correspond to porous structures with two window size distributions, narrow and wide, but the same average window diameter. The figure also includes the variation of the calculated porosity with the windowto-cell diameter ratio, for a pore structure with a coordination number of six. It can be seen that the plots for K* and e have a similar shape, but the dimensionless permeability shows a stronger dependence with dw/dc, specially for larger values. Such a strong dependence of K* is a result of the effect that the constriction window has on the cell pressure drop, as seen in Figure 10(b). Figure 11 points out that the permeability is a function of the window size distribution; a narrow distribution leads to lower values of specific permeability. This result highlights the importance of controlling the structure of the filter precursor, polyurethane foam, or pore former particles in tailoring permeabilities. Haring and Greenkorn t2s} found similar results about the effect of the radius distribution of pores on the permeability of packed beds. The window diameter distribution also affects the priming head of filters.t8,25] Figure 12 shows a comparison between predicted and experimentally determined specific permeabilities. It can be seen that the permeability of both foam and CEFILPB filters are well described by the model. The permeability of CEFILPB filters were computed according to the following equation: (d5w) l?,ef K = G(Ct)H(a) (d3)(_Ap~) [35] where the effective porosity, ~e/= e - e<, has replaced 168--VOLUME 26B, FEBRUARY 1995 I I t 10 o Porosity 0.9 I 9 Permeability 7 (narrow distribution)[ v Permeability I (wide distribution) / / v / /O /// o7 0.8 , 0.7 8 0 o 6 0.6 Q) ,7 04 4 0 /. ~ 0.5 2 0 0.1 , 0.2 0.3 ' 0.5 0.4 0.4 0.6 dw/d e Fig. 1 1 - - P l o t s showing computed dimensionless specific permeability and porosity as a function of dw/dc. The permeability plots correspond to porous structures with narrow and wide window diameter distributions. 6 e i i i - - i, ~e i 9 CEFILPB (1~e=0.52 r e k~ i ) 5 ](3 _ _ F f~i l t e r s 9 _ _ ~ 4 .,'4 "" ~ O e3 9 ID 1~ e 2 ~ el oQ) e0 e0 I I I I I eI e2 e3 e4 e5 e6 Experimental permeability, K (KD arcy) Fig. 12--Predicted vs experimentally determined specific permeabilities of foam and CEFILPB filters. the porosity, e, appearing in Eq. [31]. The effective porosity accounts for the fact that in filters of lost packed bed, there is a critical porosity, e<, obtained when the pore former spheres are just touching at one point; i.e., the resulting pores are not connected. The critical porosity value corresponds to the ordinate intercept of the curves in Figure 5. For a coordination number of six, the critical porosity is 0.52. It should be noted that the foam filters do not require such a correction since their fabrication method is completely dissimilar from the CEFILPB's. A ceramic foam filter is an open filamentary net that will keep the pore connection independently METALLURGICAL AND MATERIALS TRANSACTIONS B of the actual porosity. For this filter, no critical porosity is expected as long as this filamentary structure is not blocked by an excess of ceramic paste. Sane et al. t2~ reported experimentally determined critical porosities for foam and pore former filters, ranging from 0.62 to 0.82; however, the authors do not elaborate on the nature of ~c" The results in Figure 12 give an indirect validation of the computed laminar fluid flow field in porous media used for molten metal filtration. Additional work on this topic is being focused in the calculation of particle trajectories and filtration efficiencies under pressure drop conditions encountered in actual filtration practices. dc/2 dw/2 Xo V. CONCLUSIONS A general mathematical model has been developed to represent the fluid flow in ceramic foam filters used for molten metal filtration. The model is based on the analysis of creeping Newtonian fluid motion through a unit cell, geometric model, which resembles the pore structure of ceramic foam and pore former filters. 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 have been calculated from directly measured structural parameters; i.e., there is no need to include any fitting parameters such as tortuosity nor connectivity. It was found that the permeability of the pore former filters depends on the porosity difference e - ec rather than on the porosity e. This behavior reflects the influence of the fabrication method on the porosities that can be achieved, while maintaining interconnection between the pores. It was found that foam and CEFILPB filters have a coordination number of six. A critical porosity value of 0.52, for CEFILPB filters, was determined based on the rigid sphere model of a cubic lattice. The results indicate that the specific permeability is strongly dependent on the window size distribution. A close control of porosity and pore dimensions would allow one to tailor filter permeability values that respond to specific conditions. The intended applications of the geometric and fluid flow models developed in this work are in the study of inclusion trajectories and the evaluation of filtration efficiencies as a function of structural parameters. This work will appear in a later publication. Fig. 13--Sketch of the section of a half-pore showing the relationship between de, dw, and x0. Mean Volume of Unit Cells (Vuc) The volume of a unit cell is determined from the revolution solid formulae: Vuc = rrr~d [A21 where the radial distance, from the pore axis to the wall surface, rs, is given by the Pitagoras' theorem (Figure 14) as follows: [A31 = ~ + (x - :co)z Combining Eqs. [A2] and [A3] and integrating, results in Vtsc= ar[2r,2oc0- 2/3x 3] [A41 and substitution of Eq. [3] in Eq. [A4] leads to 7 r [ 2C~(C~ - 1 ) , / 2 _ 2 -~ (C~ - 1)3/2] d 3 Vtsc= -ff [A5] 4' .. 9 .~. 9 9 . ..... . APPENDIX Relationship between the Distance Xo and the Window Diameter dw j From Figure 13 and using the Pitagoras' theorem, we have 41- (dJ 2 ) 2 = (dw/2) 2 + ~ and recalling the definition for C~, the distance between the centers of two adjacent pores is expressed by 2Xo = d,(C~ - 1)1/2 METALLURGICAL AND MATERIALS TRANSACTIONS B ~ -- '4 [A1] [4] r " -- Xo I ' . . . . 4w X-X o 9 X Fig. 14--Sketch of a section of a pore showing the relationship between re, rs, x, and x0. VOLUME 26B, FEBRUARY 1995--169 therefore, the corresponding mean volume cell is given by <Vuc)= F(C,)(d~) [28] where, F(CO = -~ 2C~(C~ - 1) '/~ - ~ (C~ - 1Y/~ [301 aP* ~uc ~d'*c quc a LIST OF SYMBOLS laminar coefficient (Eq. [1]) turbulent coefficient (Eq. [1]) cross-sectional area of a pore at a given A x-position, total cross-sectional area of the filter Cartesian axes with origin located at the A,B pore center window cross-sectional area Aw ratio of cell-to-window diameters C1 randomly distributed cell diameter (mm) d~ maximum limit for the random diameter dc,max of a cell (mm) minimum limit for the random diameter of dc,tl~in a cell (mm) randomly distributed window diameter dw (mm) third momentum of the window diameter distribution (mm 3, Eq. [28]) fifth momentum of the window diameter distribution (mm 5, Eq. [25]) E(dw, O, q~) joint probability density function of the random variables dw, 0, and q~ fractional frequency of windows having a fd. diameter dw gravity acceleration vector g geometric function defined by Eq. [29] F(CO geometric function defined by Eq. [32] G(CO coordination number function (Eq. [33]) H(a) specific permeability (1 K Darcy = K 10 -5 cm 2) side length of a cubic unit lattice L mass concentration of inclusions in the mi metal flow at the filter inlet mass concentration of inclusions at the mo filter outlet momentum of the fluid, in a pore, in the mx direction of the main flow (Eq. [13]) momentum of the fluid, in the whole Mx filter, in the direction of the main flow (Eq. [101) coordination number of a pore N number of pores per unit volume of filter (Eq. [261) Reynolds number NRe local pressure P pressures at the inlet and at the outlet of P I , P2 the filter, respectively local pressure gradient Vp magnitude of the macroscopic pressure IVP[ gradient (KPa/m) defined as the ratio (Pl - P2)/8 al a2 170--VOLUME 26B, FEBRUARY 1995 aur r~ r~ u Us u~ [Ux]x <ux> Uw Vuc (Voc) vs X, y X X :Co Ot 8 V E Ec e~f 0 P dimensionless pressure drop in a unit cell, at Reynolds number equal to one (Eq. [20]) pressure drop through a unit cell, Eq. [7] dimensionless pressure drop through a unit cell, Eq. [19] voluminic flow rate, in a cell, in the direction of the main flow, Eq. [15] voluminic flow rate through the whole filter, cm3/s voluminic flow rate through a cell, Eq. [181 radius of a pore radial distance from the axis of a pore to its wall local value of the v-variable (pressure or velocity) in the k-iteration local fluid velocity fluid superficial velocity fluid velocity component in the direction of the pore axis projection of ux in the direction of the main flow mean fluid velocity, within the filter, in the direction of the main flow mean velocity of the fluid at a window total volume of the filter volume of a unit cell (Eq. [A6]) mean volume of the unit cells (Eq. [27]) volume of the spheres contained effectively in a unit lattice total volume of a unit lattice body-fitted coordinate axes located in a pore axis of a pore; position along such an axis axis in the direction of the main flow distance from the center of a pore to the center of its window limit value for O under which axisymmetrical flow occurs filter thickness Nabla operator filter porosity filter critical porosity filter effective porosity randomly distributed azimuthal angle fluid dynamic viscosity (Kg/~m s) fluid kinematic viscosity (/zm2/s) randomly distributed angle formed between the pore axis and the main flow direction fluid density (Kg//zm 3) ACKNOWLEDGMENTS The authors are grateful to the Mexican Council of Science and Technology (CONACyT) for support of this work. The authors express their gratitude to Dr. Arturo Palacio for his useful advice in using the PHOENICS code. METALLURGICALAND MATERIALSTRANSACTIONSB REFERENCES 1. L.S. Aubrey and J.E. 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