Cellular Ceramics / 4

4.2 Permeability 323

The fluid velocity vs in Eq. (24) is obtained by dividing volumetric flow rate by the total frontal open area exposed to flow. Alternatively, the Fanning friction factor ff can be used, considering that fd = 4 ff. Friction factors can then be determined through empirical correlations [4, 43], or charts [44] for laminar and turbulent regimes, providing that the channel Reynolds number (Rech = dhvs /l) and the roughness of cell walls are known.

The critical Reynolds number Rec for transition from laminar to turbulent flow in noncircular channels varies with channel shape. In rectangular ducts, 1900 < Rec < 2800, and in triangular ducts, 1600 < Rec < 1800 [4].

At the channel entrance, a certain distance is required for flow to adjust from upstream conditions to a fully developed flow pattern. This distance depends on the Reynolds number Rech and on the upstream flow conditions. For a uniform velocity profile at the channel entrance, the computed length in laminar flow required for

The frictional losses at channel entrance are larger than those of similar length for fully developed flow. At the channel exit, the velocity profile also undergoes rearrangement, but the exit length is much shorter than the entrance length. At low Rech, it is about one channel radius. At Rech > 100, the exit length is essentially zero.

If the honeycomb channels are shorter than Le, then Eq. (24) should no longer be applied. In this case, the pressure drop can be roughly estimated on the basis of the theory of flow through perforated plates (thick cell walls) or through screens (thin

where, vs is the superficial velocity based upon the total frontal area of the plate or screen of orifice/opening dh, and j the fraction of frontal area open for flow. The discharge (orifice) coefficient C is given as a function of the plate or screen Reynolds number Res = dhvs /jl), the hole pitch (center-to-center distance), and the thick- ness-to-diameter ratio. Y is the expansion factor for gases (for liquids, Y = 1). Charts and equations for C and Y are given in the literature [4, 45]. Studies on the fluid dynamics of different types of ceramic honeycombs are also available [40, 46–54].

Ceramic Foams

Foams are cellular structures composed of a three-dimensional packing of hollow polyhedra (cells) with edges randomly oriented in space. Depending on the application, foams can be tailored from different materials (polymers, metals, and ceramics) and with either open or closed cells, displaying different permeabilities. Most

324 Part 4 Properties

open-cell ceramic foams for fluid-flow applications are produced by two techniques which are well-described in the literature: ceramic replication of an organic substrate and foaming of ceramic slurries [55–57].

Ceramic replicas, commonly referred as reticulated ceramics, are produced from different raw materials (alumina, zirconia, silicon carbide, silica), display porosities varying from 70 % to greater than 90 %, with open tetrakaidecahedral cells of size from less than 100 mm to greater than 5 mm. Their geometrical features are determined by the polymeric sponge matrix, which is quoted in pores per linear inch (commercially available from 3 to more than 100 ppi). Depending on the application, finished elements can be shaped as disks, tubes, rods, rings, or in other cus- tom-designed configurations.

Reticulated ceramics are primarily used in molten-metal purification, but their range of applications has recently spread to many other solid–fluid contact processes, such as catalytic combustion, hot-gas cleaning, gas combustion, and heat transfer. Contrary to honeycomb monoliths, reticulated ceramics have extensive pore tortuosity and flow patterns with high degrees of lateral mixing, which favor particle–filter contact and enhance trapping and conversion efficiencies in catalytic and filtration processes.

Foamed ceramics have porosities between 40 and 90 %, with closed or open nearly spherical cells, and pore sizes ranging from less than 10 mm to about 2 mm. They can be produced from several ceramic oxides (for filtration applications) or from high-purity materials such as hydroxyapatite and calcium phosphate for a variety of applications in the biomedical field, including materials for bone repair [58], carriers for controlled drug-delivery systems, and matrices for tissue engineering [59]. They are also used in ion-exchange processes and as filters for contaminated water [56].

Despite the importance of ceramic foams for fluid-flow applications, only in recent years has their permeability been addressed consistently in the literature. Hence, experimental data relating the permeability constants k1 and k2 to foam structure are still too scarce for a comprehensive analysis.

Permeability is expected to change according to several structural features of ceramic foams (pore count, porosity, cell size, strut thickness) that reflect different aspects of fluid–solid interaction.

The most common way to label a ceramic replica has been through its nominal pore count, and for this reason this quantity has been often related to permeability. In general, an increase in pore count results in a decrease in permeability, as it indirectly infers an increase in the number of cell boundaries (struts) per unit length and thus in the frictional area.

In Fig. 2, data gathered from the literature show that permeability constants of ceramic replicas vary by about three orders of magnitude over the commercial range of pore count (8–100 ppi). Interestingly, in this pore-count interval no defined trend is observed for porosity, which fluctuates between 0.75 and 0.95. Scattering in permeability values is relatively high for similar pore counts and can be explained on the basis of the following factors: 1) the nominal pore count is not a precise measure and may vary according to organic-foam manufacturer, 2) measuring techniques for structural parameters vary from author to author, 3) the compressibility effect on

dpore (mm)

Fig. 5 Pore size dependence of properties for reticulated ceramics gor two methods to produce the cellular structure (data from Salvini et al. [64]).

Most models used in prediction of permeability parameters for ceramic foams are derived from correlations originally developed for granular beds, and this often leads to inaccuracies [5]. The major difficulty in derivation of such models is identifying a characteristic length that represents the cellular media realistically and thus replaces the particle size dp in the Ergun-like equations described in Section 4.2.4.1. The most obvious approach involves use of an equivalent pore size dc, usually identified as an equivalent circle diameter obtained, for instance, by image analysis of foam specimens. Philipse and Schram [62] proposed that dp could be directly replaced by dc in the Ergun equation and found experimentally that the k1/k2 ratio is roughly proportional to dc in 20–90 ppi alumina–silica foams.

Another approach suggests that dc could be represented by the cylindrical form of the hydraulic diameter, related to dp by

Permeability constants k1 and k2 based on Ergun equations (13) and (14) can then be rewritten in terms of dc respectively as:

The numerical values in Eqs. (29) and (30) are based on granular beds and may be replaced by values that can be experimentally determined for cellular materials. Such an approach was investigated with SiC–Al2O3 ceramic replicas from 30 to 75 ppi [65]. Nevertheless, results showed that the accuracy of Eqs. (29) and (30) depends remarkably on the thickness of the foam specimens used for pore size evaluation.

A more sophisticated correlation for predicting permeability of foams has been proposed for metallic compositions in which the cellular medium is modeled as cubic unit cells, leading to the following correlations [63]:

where s is the tortuosity of the porous matrix, that is, the mean distance covered by the fluid over the thickness of the crossed porous medium. The tortuosity can be measured in electric resistivity experiments or estimated in terms of the medium porosity e by:

In another recent model, proposed by Richardson et al. [61], the pressure drop values for 10 to 65 ppi reticulated foams made of Al2O3 and of ZrO2 were associated to Ergun equations, resulting in the following expressions for k1 and k2:

Three different models were examined for calculation of the specific surface area So, including one that is based on comprehensive work by Gibson and Ashby [66], in which the foam is treated as a regular packing of tetrakaidekahedra. Surprisingly, it has been found that the “hydraulic diameter model” proposed by Kozeny for packed beds and based on the pore diameter dc, is sufficient to fit experimental data:

332 Part 4 Properties

where d is the pipe diameter, and v the average fluid velocity. For Re < 2000 the flow regime is said to be laminar, and for Re > 2200 it is characterized as turbulent.

Several authors [35, 71, 72] have proposed variations on the Reynolds number to take into account microscopic structural features which would enable characterization of the flow regime in porous media. Commonly, the tube diameter d in Eq. (41) has been replaced by a characteristic length of the pore structure d, which can be considered as the equivalent particle diameter or the hydraulic pore diameter for granular beds [14, 35, 73], the strut diameter or the cell size for cellular materials [62, 71, 72, 74, 75], or the fiber length for fibrous media [31]. Similarly, the superficial velocity vs defined earlier has sometimes been replaced by the pore, or interstitial velocity vi. The lack of a standardized definition for these parameters has resulted in conflicting values for the transition between viscous and inertial regimes in the literature.

According to one of the approaches [73], an interstitial Reynolds number Rei is defined as:

where dc is the equivalent pore diameter and e is the porosity of the medium.

For Rei < 1, the energy is dissipated into the porous medium only as a result of viscous friction between the fluid layers near the pore walls. The pressure drop in this case is linearly proportional to the fluid velocity, and Darcy’s law is valid.

The key assumption that underlies Darcy’s law is that the Reynolds number is considered to be sufficiently small that fluid inertia effectively plays no role in the dynamics of flow. In practice, Darcy’s law shows good agreement with experiment for a wide range of flow, but it breaks down at Rei » 1–10 due to the onset of inertial forces within the laminar regime, and not due to the onset of turbulence, as commonly thought [32, 35].

For Rei > 1–10, inertial effects initiate in the laminar regime, with the emergence of disturbances in the streamlines owing to the curvature of flow channels. This additional flow resistance would explain the nonlinear relationship between pressure drop and fluid velocity represented by the term vs2/k2 in Forchheimer’s equation, which holds until Rei » 150.

For 150 < Rei < 300, an unsteady flow regime occurs, and for Rei > 300 a highly chaotic flow regime is observed, which might be associated with true turbulence, as occurs in straight tubes.

The transition from laminar to turbulent flow in reticulated rigid foams was investigated by Seguin et al. [71, 72], based on the pore Reynolds number (Rep =vsdc/l). Some representative data of their study are shown in Table 4.