Cellular Ceramics / 4
.2.pdf
4.2 Permeability 333
Table 4 Onset of turbulence in reticulated rigid foams. Adapted from Seguin et al. [71, 72].
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Nominal pore count |
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10 ppi |
20 ppi |
45 ppi |
100 ppi |
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Rei transition (Eq. 42) |
390 |
470 |
>257 |
>82 |
% inertial effect on pressure drop at transition |
83 |
85 |
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Re transition (Eq. 41) |
4350 |
8050 |
>11 000 |
>11 000 |
Interestingly, these authors verified that laminar flow in tubes filled with reticulated foams extends far above the limit of Re » 2000–2200 established for empty tubes. The “laminarizing effect”, increases with decreasing pore size. A similar effect has been observed in the literature for ceramic foams with 4, 8, and 12 pores per centimeter [74–75].
The difficulty in evaluating the pore Reynolds number, due to the complexity in establishing a representative characteristic pore length d, motivated some authors to develop alternative methods for determining the flow regime and thus verifing the applicability range for Darcy’s law.
A criterion based on the velocity vs, fluid properties l and , and features of the porous structure L, k1, and k2 was proposed by Ruth and Ma [76] to determine the relative contribution of viscous and inertial resistances and to decide whether Darcy’s law or Forchheimer’s equation is the best permeability equation for a given application.
For this purpose, the dimensionless Forchheimer number Fo was defined by
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Fo ¼ |
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The Fo parameter represents the ratio between kinetic and viscous forces that contribute to fluid pressure drop. Since the ratio k1/k2 is expressed as length, Fo can be understood as an analogue of the pore Reynolds number, with d = k1/k2. Fo is related to the linearity in the pressure drop curve in the same way that Rei is related to the laminarity of flow. Based on the Fo parameter, Forchheimer’s equation (5) can be rewritten as:
DP |
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L |
k1 |
The individual contribution of viscous and inertial resistances on the total pressure drop can now be estimated through:
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DPtotal |
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1þFo |
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DPinertial |
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Fo |
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DP |
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334 Part 4 Properties
Figure 8 shows the contributions of inertial and viscous resistances to the total pressure drop based on the Forchheimer number. For Fo << 1, inertial effects are negligible (DPtotal » DPviscous) and Eq. (10) reduces to Darcy’s law. On the other hand, when Fo >> 1, viscous effects can be disregarded (DPtotal » DPinertial) and the pressure drop can be reasonably estimated through:
DP |
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For any other intermediary flow condition both terms are relevant and the complete Forchheimer’s equation should be taken as primary permeability model. For compressible fluids, such as gases and vapors, the term DP in Eqs. (44)–(47) should be replaced by (Pi2–Po2)/2 PL.
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Fo/[1+Fo] |
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∆Pinertial/∆Ptotal = |
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∆Pviscous/∆Ptotal |
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Forcheimer number, Fo (-)
Fig. 8 Percentage of viscous and inertial pressure drops according to the Forcheimer number.
Despite the previous discussion, Darcy’s law has been still preferred to Forchheimer’s as the permeability equation in several engineering applications involving porous materials. This is either due to the proposed linearity between DP and vs, which makes the inclusion of the law in mathematical modeling easier, or because the fluid velocity chosen for the desired application is low enough to disregard inertial effects. The abundance of studies dealing with Darcy’s law has unfortunately spread its use in applications involving porous ceramics, frequently without previous analysis of its validity. The examples that follow illustrate the practical importance of choosing the appropriate permeability equation and the applicability of the Forchheimer number to help in the choice.
4.2 Permeability 335
Example 1. Consider a 20 ppi SiC ceramic foam filter with 5.9 cm diameter, 2.14 cm thickness, and permeability constants k1 = 2.36 0 10–8 m2 and k2 = 1.85 0 10–3 m. Determine the column head required to achieve a vertical fluid velocity of 0.10 m s–1 through the clean filter where the fluid is a molten metal ( mm = 8000 kg/m3, lmm = 2.0 0 10–3 Pa s). Compare the adequacy of Darcy’s law and Forchheimer’s equation.
Resolution:
a) Darcy’s law for incompressible fluids: |
DP |
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lmm |
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L |
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DP |
2:00 · 10 3 Pa:s |
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ð0:10 m=sÞ fi DP = 181.3 Pa. |
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0:0214m |
2:36 · 10 8 m2 |
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The column head DHmetal is given by DH ¼ |
DP |
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mm g |
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fi DHmetal ¼ |
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fi DHmetal = 2.31 mm. |
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ð8000kg=m3 Þð9:8m=s2 Þ |
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b)Forchheimer’s equation for incompressible fluids:
DP |
lmm |
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2 |
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vs þ |
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vs , |
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L |
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DP |
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2:0 · 10 3 Pa:s |
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8000 kg=m3 |
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fi DP = 1106.8 Pa. |
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0:0214 m |
2:36 · 10 8 m2 |
1:85 · 10 3 m |
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DHmetal ¼ |
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1106:8 Pa |
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fi DHmetal = 14.1 mm. |
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ð8000 kg=m3 Þð9:8 m=s2 Þ |
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Table 5 Summary of data for Example 1.
Fluid |
Darcy’s law |
Forchheimer’s |
Fo |
DPviscous |
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DPinertial |
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DP |
DP |
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total |
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total |
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DP/Pa DH/mm |
DP/Pa DH/mm |
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Molten metal |
181.3 |
2.3 |
1106.8 |
14.1 |
5.10 |
16.4 |
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As seen in Table 5, Darcy’s law predicted a pressure drop that is only 16.4 % of that given by Forchheimer’s equation. In fact, if this column head of 2.3 mm were used, the actual velocity through the filter would be 3.6 cm s–1 and not 10 cm s–1 as desired. Under this flow condition (Fo = 5.1), the inertial resistance contributes more than 83 % of the total pressure drop, mainly because of the high fluid density. Thus, Darcy’s law does not hold for this application, and Forchheimer’s equation should be used to predict the flow behavior of metal through the foam.
Interestingly, if the fluid were air at 25 C (l = 1.83 0 10–5 Pa s, = 1.18 kg m–3) flowing at the same velocity, then Fo = 0.08 and viscous effects would represent 98 % of the total pressure drop, validating Darcy’s law. However, even if air-flow tests were used to assess permeability data, as commonly found in certain foam catalo-
336 Part 4 Properties
gues, Darcian-like information of the type “vs for a given DP” should be avoided, since it does not allow reliable prediction of inertial pressure drop for liquid-flow applications such as in foundries and steelmaking plants.
Example 2. A set of rigid fibrous alumina filters is to be used in a incineration facility for toxic waste to clean the exhaust gas stream at a temperature of 800 C and absolute inlet pressure of 1 atm (1.013 0 105 Pa). Determine the operational filtration velocity through a single clean candle (L = 1.63 cm, k1 = 6.27 0 10–11 m2, k2 = 4.49 0 10–6 m) if the maximum permissible pressure drop Pi–Po is 150 mmH2O
(1500 Pa). Consider that fluid properties and velocity are based on the filter entrance ( gas = 0.33 kg m–3, lgas = 5.71 0 10–5 Pa s).
Resolution:
Po = Pi–1500 Pa = (1.013 0 105–1500) Pa fi Po = 0.998 0 105 Pa
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Pi |
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a) Darcy’s law for compressible fluids: |
2PL |
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Then |
ð1:013 · 105 PaÞ2 ð0:998 · 105 PaÞ2 |
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5:71 · 10 5 Pa:s |
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fi vs = 10.04 cm s–1. |
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b) Forchheimer’s equation for compressible fluids: |
Pi |
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2Pi L |
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ð1:013 · 105 PaÞ2 ð0:998 · 105 PaÞ2 |
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5:71 · 10 5 Pa:s |
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0:33kg=m3 |
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6:27 · 10 11 m2 |
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fi vs = 9.98 cm s–1.
Table 6 Summary of data for Example 2.
Fluid |
DP = Pi-Po |
Darcy’s law |
Forchheimer’s |
Fo |
DPviscous |
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vs/cm s–1 |
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Air |
1500 |
10.04 |
9.98 |
0.008 |
99.2 |
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Under this flow condition (Fo = 0.008), there is almost no contribution of inertial effects (0.8 %) to filter pressure drop, and both permeability equations converged to the same velocity value (Table 6). Thus, even when considering occasional fluctuations in vs, P, T, or k1 and k2 values (due to pore clogging), Darcy’s law could be safely used to model the filter air flow in such an incineration facility.
From these two examples, it becomes clear that the concept of high and low velocity in flow through a porous medium is not absolute and should not be used as the only criterion to validate Darcy’s law. Although both ceramic foam and fibrous filter operated at the same fluid velocity, in the former case flow was mostly inertial, whereas in the latter it was purely viscous.
4.2 Permeability 337
The use of Fo to determine the validity of each permeability equation therefore implies that both constants k1 and k2 must be available for analysis, which is anyway easier to accomplish experimentally than evaluating a characteristic length d to use in the pore Reynolds number Rei.
Recently, a substantial volume of permeability data based on Forchheimer’s equation for highly porous media has become available in the literature. A compilation of representative experimental k1 and k2 values for several porous materials is shown in Fig. 9. Table 7 classifies these materials according to their location in Fig. 9.
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[68] |
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-0.08093 |
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permeability, |
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Fig. 9 Permeability data gathered from the literature for several porous materials. # Data for metallic foams.
Table 7 Classification of porous materials based on their location in Fig. 9.
Material |
Location in Fig. 9 |
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Green refractory castables, mortars |
A1–A3 |
Fired castables, bricks |
B3–B5 |
Starch-containing ceramics |
B3–B4 |
Gel-cast foams |
B3–B4–C4–C5–D6–E7 |
Fibrous filters |
C6–D6 |
Honeycomb (wall-flow) filters |
C5–C6 |
Granular beds (spherical particles) |
D6–D7 |
Metallic reticulated foams (45–100 ppi) |
E7 |
Ceramic reticulated foams (8–90 ppi) |
E6–E7–F8 |
338 Part 4 Properties
Despite all the different techniques, fluids, and flow conditions used to assess permeability data, it is remarkable that a clear correlation between both permeability parameters could link porous media of totally different structural features. The best fit (r2 = 0.9823) considering the whole data set with 640 points was:
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where units of k1 and k2 are respectively m2 and m.
Extensive literature data involving mainly air-flow applications could not be included in the map of Fig. 9 because only k1 values were available. In such cases, however, given the apparent success of the proposed fitting, Eq. (48) could be used to find approximate values for the non-Darcian permeability k2 and thus estimate the inertial contribution to flow resistance for a desired application.
4.2.6
Summary
A three-dimensional structure that combines a high degree of porosity and a controlled size range of interconnected pores makes cellular ceramics some of the most permeable rigid porous media currently used in engineering applications. Nevertheless, reliable evaluation of permeability parameters for these materials is recent and has faced different challenges.
The difficulty of defining a fluid-dynamic characteristic length for the cellular geometry has been the major obstacle for reliable permeability modeling. On the other hand, recent equations to predict permeability constants based on the cellular structure have not been extensively tested, and the influences of parameters such as pore count, pore size, porosity, and strut thickness are not yet well established.
Attempts to fit fluid flow data according to Darcy’s law have also caused inaccuracies, since inertial effects generally prevail in pressure drops. The best approach currently available to assess permeability of cellular ceramics is still by fitting experimental data to Forchheimer’s equation, in which both Darcian (k1) and non-Darcian (k2) permeability constants are obtained. The combined analysis of fluid and flow conditions and the use of Forchheimer’s number Fo will then determine the most representative permeability parameter for a given application.
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4.2 Permeability |
339 |
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References |
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1 |
Innocentini, M.D.M., Pardo, A.R.F., |
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Salvini, V.R. and Pandolfelli, V.C., Am. Ceram. |
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Innocentini, M.D.M., Salvini, V.R., Coury, J.R. |
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1999, 82[7], 1945–1948. |
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guese). |
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Macdonald, I.F., El-Sayed, M.S. and Dullien, |
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Seville, J.P.K., Clift, R., Withers, C.J. and |
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