Cellular Ceramics / 4

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4.2

Permeability

Murilo Daniel de Mello Innocentini, Pilar Sepulveda, and Fernando dos Santos Ortega

4.2.1

Introduction

Fluid flow through cellular ceramics is a key topic in many engineering applications. Due to their unique pore geometry that provides very low pressure drop and the ability to operate in extreme conditions, cellular ceramics are the ideal substitutes for conventional porous media used in high-temperature processes involving filtration, fluid mixing, chemical reaction, or mass transfer. As the efficiency of these processes depends primarily on the permeability of the porous media employed, attention in this chapter is drawn to concepts of permeability evaluation and modeling of porous ceramics.

Quantification of permeability is initially discussed on the basis of the equations most applied in the literature. Concepts of flow through granular and fibrous media are then introduced, as they compose the basis of most correlations currently applied to cellular ceramics. Following this, experimental permeability data for honeycomb and foamlike ceramics gathered from the literature are discussed and related to structural features such as porosity, cell size, and, whenever possible, to processing variables. The most recent and innovative models for predicting permeability parameters of cellular ceramics are described and, in cases when no reliable modeling is possible, the reader is referred to other well-established theories available in the literature. The final section concerns criteria and practical examples to establish the correct flow regime through porous media and thus help to choose an optimum permeability model for a given application.

4.2.2

Description of Permeability

Permeability is regarded as a macroscopic measure of the ease with which a fluid driven by a pressure gradient flows through the voids of a porous medium. Thus, permeability is neither a property of the fluid nor a property of the porous medium, but reflects the effectiveness of the interaction between them [1].

Permeability of a porous medium is commonly expressed through parameters that are derived from two main equations: Darcy’s law, Eq. (1), and Forchheimer’s equation, Eq. (2),

Cellular Ceramics: Structure, Manufacturing, Properties and Applications.

Michael Scheffler, Paolo Colombo (Eds.)

Copyright 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-31320-6

where –dP/dx is the pressure gradient along the flow direction, l and are respectively the absolute viscosity and the density of the fluid, and vs is the superficial fluid velocity, defined by vs = Q/A, where Q is the volumetric flow rate and A is the exposed surface area of the porous medium perpendicular to flow direction, and k1 and k2 are usually known as Darcian and non-Darcian permeabilities. These parameters incorporate only the structural features of the porous medium and therefore are considered constant even if the fluid or the flow conditions are changed. Note that although both parameters are referred to as “permeabilities”, they are dimen-

sionally distinct: k1 is expressed in dimensions of square length [e.g., m2, perm (10–4 m2), and Darcy (10–12 m2)], whereas k2 is expressed in dimensions of length

(typically m).

The basic difference between Eqs. (1) and (2) is the type of dependence that the fluid pressure assumes when related to the fluid velocity. Darcy’s law, which is derived from experiments conducted at very low velocities, considers only the viscous effects on the fluid pressure drop and establishes a linear dependence between the pressure gradient and the fluid velocity through the porous medium. Forchheimer’s equation, on the other hand, considers that the pressure gradient displays a parabolic trend with an increase in fluid velocity due to the contributions of inertia and turbulence. Although Darcy’s law has been widely applied in the literature because of its simplicity, Forchheimer’s equation has yielded more realistic and more reliable permeability parameters [2]. In fact, accurate use of Darcy’s law and Forchheimer’s equation depends very much on a correct determination of the flow regime within porous media. This controversial topic is discussed further in Section 4.2.5.

The integration of the pressure gradient –dP/dx through the porous medium depends on the fluid compressibility. In contrast to liquids, gases and vapors expand along the flow path, and thus fluid velocity at the exit of the porous medium becomes higher than at the entrance, and this affects the pressure drop profile. This effect can be taken into account in Eqs. (1) and (2) by considering the fluid as an ideal gas. Table 1 presents integrated forms of Darcy’s law and Forchheimer’s equation according to the fluid compressibility.

The compressibility effect is more pronounced for low-porosity materials, such as rocks, bricks, tiles, concrete, and cast refractories, but it may also be considerable for highly porous media, depending on the thickness of the sample and on the pressure or velocity applied. Thus, permeability assessment by gas-flow experiments is more likely to generate errors if the compressibility effect is neglected [3].

* Where: DP = Pi–Po, L = medium thickness along the flow direction,

Pi = absolute fluid pressure at the medium inlet, Po = absolute fluid pressure at the medium outlet, P = absolute fluid pressure for which

vs, l, and are measured or calculated (usually Pi or Po).

4.2.3

Experimental Evaluation of Permeability

Experimental evaluation of permeability parameters consists of a test in which a fluid is forced to flow through a porous sample in stationary regime. The sample is generally a disk or cylinder of thickness L and exposed area A, laterally sealed between two chambers. Absolute fluid pressure at the inlet (Pi) and outlet (Po) of the sample are measured and recorded as a function of flow rate Q or fluid velocity (vs = Q/A). If Forchheimer’s equation is used, the collected data set is fitted according to the least-squares method to a parabolic model of the type: y= ax + bx2, where y is either Pi–Po for liquids or (Pi2–Po2)/2 PL for gases and vapors, and x is the velocity vs. For gas flow, velocity should be also corrected for the actual average temperature and pressure through the sample. The permeability parameters are then calculated from the fitted constants a and b, respectively, by k1 = l/a and k2 = /b. Table 2 provides equations that are useful for estimating the density and the viscosity l of water or common gases according to test temperature and pressure.

Several apparatuses for permeability evaluation are described in the literature [5]. The best configuration will depend mostly on the features of the sample. Figure 1 illustrates two typical setups used for ceramic foams. For highly porous materials, liquids are preferred as the testing fluid to provide a higher contribution of the inertial term ( vs2/k2) and therefore reliable fitting of k2 values. For consistent data analysis, samples from different batches should have their permeability parameters fitted within a similar pressure–velocity range [6]. If the sample is anisotropic, attention should be given to the flow direction. Generally, the experiment must be carried out with the sample mounted in the same direction as its regular usage.

316Part 4 Properties

Table 2 Useful equations for estimating the density and viscosity of water and common gases at different temperatures and pressures.*

* Eq. (7) is a polynomial fitting of data given in Ref. [4]. In Eqs. (7)–(10), T is given in C and P in Pascal for and l in SI units. R is the ideal gas constant (8.314 J mol–1 K–1).

318 Part 4 Properties

utilizing the complete Navier–Stokes equation; and 3) models based on flow around submerged objects.

The complexity of flow patterns and the difficulty to mathematically describe the pore structure have precluded the use of the aforementioned approaches to reliably predict permeability parameters for a general porous medium. For engineering purposes, valuable permeability modeling is available in the literature only for special classes of porous media, such as unconsolidated granular media (sand filters, fixed beds, etc.), consolidated granular media (ceramic filters, polymer membranes, bricks, concretes, mortars, etc.), and fibrous media (bag, candle, and cartridge filters). The permeability of cellular ceramics has been only recently addressed in the literature, and in many cases, models used in the determination of permeability constants are still derived from correlations originally developed for other classes of porous media.

The aim of this section is to give a brief overview of the main useful equations relating the permeability parameters to the different types of porous structures, as applied in the literature.

4.2.4.1

Granular Media

Most correlations currently applied to predict permeability of porous ceramics were developed in the 1900s on the basis of data for flow through unconsolidated granular media (i.e., a bed of loose particles) obtained for viscous flow [8–10] and inertial flow [11–13].

The Kozeny–Carman equation stems from the Hagen–Poiseuille relationship (an exact analytical solution for viscous flow in a capillary tube) and gives satisfactory predictions for unconsolidated porous media. Based on the Kozeny–Carman equation, the Darcian permeability k1 defined in Eqs. (1) and (2) is written as

where e is the volumetric void fraction (porosity) of the bed, Kk the Kozeny parameter, and So the specific surface area, that is, the surface area of solid phase exposed to the fluid per unit volume of solid phase. So is usually measured by the Brunauer–Emmett–Teller (BET) method. The value of the Kozeny parameter is a function of the structure of the medium, and similar channel shapes will have the same value of Kk over a range of values of e and So. For spherical particles, Kk is equal to 4.8 and for “irregular particles” it is around 5.0.

The permeability constant k1 is related to an equivalent spherical particle diameter

Ergun [14], working with experimental results gathered in both viscous and inertial regimes, developed the following relationships to describe k1 and k2 for packed

4.2 Permeability 319

columns made of spheres, cylinders, tablets, nodules, round sand, and crushed materials (glass, coke, coal, etc.):

where dp is the equivalent particle diameter given by Eq. (12).

Macdonald et al. [7] modified the constants of the Ergun equation by fitting a larger number of experimental data and considering that, besides Eq. (12), the equiva-

where dv is the average volumetric diameter and u is the particle sphericity. The relationships found for k1 and k2 are:

For particles with intermediary roughness, the numerical value in Eqs. (17) and (18) would lie between 1.8 and 4.0. Pressure-drop predictions across beds of coarser granular particles (dp > 50 to 100 mm) from these equations should give an accuracy of about 50 % and hold over a wide porosity range (0.36 < e < 0.92) [7].

Despite criticisms, Ergun-like relationships are widely used at present to predict fluid flow through consolidated and unconsolidated porous media in several fields, including groundwater hydrology, petroleum engineering, water purification, industrial filtration, and concrete infiltration. Many works have tried to establish similar correlations, usually of empirical type and applicable to a single class of porous materials; for instance, for a bimodal grain size distribution [15], for a bed of particles which are themselves very porous [16], for consolidated porous media that generally exhibit high tortuosity and for those having large or bimodal pore or grain size distribution [17, 18], and for highly porous anisotropic consolidated media [19]. Lin and Kellett [20] described several permeability equations for powder compacts, and Innocentini et al. [21] related k1 and k2 constants to pore size and porosity for consolidated refractory ceramics. Comprehensive theoretical and empirical reviews on fluid flow through granular media are also available in the literature [22–25].

320Part 4 Properties

4.2.4.2

Fibrous Media

Fluid-dynamic applications of fibrous media are essentially related to filtration processes of liquid or gaseous suspensions and more recently to heat-transfer devices. Fibrous media can be classified into woven, nonwoven (felted), and sintered media, and are shaped as candles, coupons, bags, or cartridges. For room-temperature applications, fibers are made from paper, cotton, wool, or synthetic materials (polyamide, polyester, polyacrylonitrile, polyethylene). For filtration at higher temperatures, fibers are made of special polymers (Nomex, Teflon), metals or metal alloys (copper, bronze, stainless steel), and mostly of ceramics (alumina, mullite, zirconia, silica). In this case, the fiber diameter usually varies between 2 and 20 mm, and porosity of the medium varies according to the fabrication method: 0.35 < e < 0.5 for woven, and

0.6 < e < 0.95 for felted media.

Typical fluid velocity in aerosol filtration is lower than 10 cm s–1, and given the low gas density, inertial flow resistance has been often omitted from the analysis.

For this reason, mostly Darcian permeability (k1) data are found in the literature, varying from 10–15 to 10–10 m2 [26, 27].

Seville et al. [28] related the permeability to the fiber structure of ceramic elements of very high porosity (e » 0.8–0.95) and specific surface area [So » (0.8–1.5) 0 106 m2 m–3] through the Kozeny–Carman equation (Eq. (12)), in which the Kozeny constant Kk was experimentally found to be around 6.0. Innocentini [27] worked with air flow through a similar commercial fibrous medium (e » 0.8) and determined average experimental values for k1 and k2 as 1.72 0 10–11 m2 and 2.04 0 10–6 m, respectively.

MacGregor [29] modified the Kozeny–Carman equation for a textile assembly to model the flow of dyes through textile yarn packages, providing a method to predict the Darcian permeability based on fiber diameter df and fabric porosity e:

Kk is around 5.5 [30], but this value can vary widely with the porosity of the medium [26].

Another frequently applied semi-empirical equation for Darcian air flow through random fiber media was proposed by Davies [31]:

This equation is valid for the range 0.6 < e < 1.0, with Ref = vsdf/l < 1.

A correlation for fiber mats with porosities greater than 0.98, based on measurements on wool, cotton, rayon, glass, and steel-wool pads, in which the fiber size varied from 0.8 to 40 mm is given by [31]:

4.2 Permeability 321

Other permeability equations are described in the literature for felted and woven filtration fabrics [26, 32], and for deformable porous media [33, 34]. Theoretically derived permeability functions were discussed in refs. [35–38], while the flow patterns through fibrous media were recently reviewed in Ref. [39].

4.2.4.3

Cellular Media

Honeycombs

Honeycombs are cellular materials with a regular arrangement of parallel identical cells (channels) employed in a variety of fluid-flow applications: automotive emis- sion-control catalysis, ozone-abatement catalysis, woodstove combustion, catalytic combustion, heat-exchange devices, molten-metal filtration, ultrafiltration, and diesel particulate filtration [40].

The honeycomb elements are pressed or extruded as cylinders and blocks (for gas flow) or tiles and disks (for liquid flow), which can be assembled in larger units to match the flow-area requirements. They are fabricated from metals (aluminum, stainless steel, special alloys) and sintered porous ceramics (cordierite, alumina, mullite, silicon carbide, zeolite, zirconia, spinel) to meet heat-transfer requirements or to withstand aggressive environments. The main processing variables are cell shape (triangular, square, hexagonal, or circular), wall thickness (0.2–1.5 mm), cell density (4–224 cells per cm2), and percentage of open frontal area (30 to over 90 %). Porosity of the ceramic walls varies from less than 0.3 to over 0.5, with mean pore sizes of 4–50 mm.

Depending on the application, ceramic honeycombs can operate in different flow configurations. Those used as particulate traps are commonly referred as wall-flow, Z-flow or dead-end filters, because the flow path is achieved by sealing half of the channels at the upstream end, in an alternating checkerboard manner, and the other half at the downstream end. Thus, particle-laden gas entering the upstream channels is constrained to flow through the porous walls, which act as surface filters, and exit through the downstream cells. On the other hand, honeycombs used in catalytic conversion or in molten-metal filtration operate in a parallel flow mode, with all cells open at both ends.

The pressure drop through clean wall-flow honeycombs can be modeled on the basis of three different contributions: the pressure drop across the porous walls DPd, the friction pressure drop along the passageways DPf, and the dynamic head pressure drop DPh due to changes in flow area within the filtering element. Under optimal filtering conditions, DPf and DPh are minimized and the total pressure drop is mainly due to DPd [41].

The flow through the porous walls is typically a case of flow through media treated in Sections 4.2.4.1 and 4.2.4.2. Since the operational pressure drop is low and the fluid velocity is normally below 20 cm s–1, DPd has been used in the literature with Darcy’s law, in which k1 is experimentally determined or predicted by Eqs. (13) or (16) for walls made of sintered particle grains, or by Eqs. (19)–(21) for bonded fibrous media. The Kozeny–Carman equation also can be used, provided that there

322 Part 4 Properties

is an appropriate value for the Kozeny constant Kk. It must be borne in mind that the superficial velocity used in DPd calculations is given by the volumetric flow rate arriving at the honeycomb element divided by the total internal face area of channels, and L is the thickness of an individual channel wall. Experimental k1 values

were determined for clean wall ceramics made of cordierite (k1 = 4.3 0 10–13 m2) and SiC (k1 = 2.1 0 10–12 m2) [42].

Relations for DPf and DPh in square cells of clean honeycomb elements were proposed as [41]:

where Lf is the filter or channel length, W the channel width, and cf a friction coefficient dependent of the wall roughness and the Reynolds number.

Reliable evaluation of the total pressure drop through clean parallel-flow honeycombs must consider the flow resistance offered by the frontal solid area (friction and border effects) and along the channels (friction and turbulence). Simplifications can be made for some particular cases of honeycomb geometry, which lead to wellknown theories of flow in channels, as described in the literature [4] and given in Table 3.

Table 3 Simplified models to estimate pressure drop through parallel-flow honeycombs.

For the case of honeycombs with medium or long channels, flow resistance is essentially due to viscous friction and turbulence inside the parallel channels. If border (entrance and exit) effects are disregarded, then pressure drop is estimated through the Darcy–Weisbach equation [4]:

where fd is the Darcy friction factor and dh is the hydraulic diameter, defined for a single channel as four times its cross-sectional area exposed to flow divided by its sectional wetted perimeter. For circular channels, dh is the cell diameter, and for square channels it is the cell width.