Mass transfer and pressure drop in ceramic foams: A description for different pore sizes and porosities

Chemical Engineering Science 63 (2008) 5202 -- 5217

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Mass transfer and pressure drop in ceramic foams: A description for different pore sizes and porosities G. Incera Garrido, F.C. Patcas ∗ , S. Lang, B. Kraushaar-Czarnetzki Institute of Chemical Process Engineering CVT, University of Karlsruhe, Kaiserstrasse 12, D-76128 Karlsruhe, Germany

A R T I C L E

I N F O

Article history: Received 10 December 2007 Received in revised form 12 May 2008 Accepted 23 June 2008 Available online 25 June 2008 Keywords: Foam Sponge Catalyst carrier Mass transfer Pressure drop Correlation

A B S T R A C T

Mass transfer and pressure drop properties of alumina open-cell foams with pore counts between 10 and 45 PPI and porosities between 75% and 85% were studied in connection with their morphology. A combination of microscopic imaging, mercury porosimetry and magnetic resonance imaging allowed the determination of the pore sizes, strut diameters, void fractions and geometric surface areas of the foams. The mass transfer coefficients of the foams were measured by monitoring the CO oxidation over Pt/SnO2 coated foams in the temperature regime where external mass transfer is rate determining. The dimensionless correlation Sh = A · ReB · Sc1/3 showed a systematic variation of A and B parameters with the foam pore size and porosity. The observed anisotropy of the foam structures required the implementation of an additional geometrical factor to obtain a unifying description of the dimensionless mass transfer coefficients 0.58 · 0.44 . for foams with different pore densities and void fractions: Shfoam = 1.00 · Re0.47 · Sc1/3 · (Dp /0.001m) h This equation was also applicable to a metallic foam (40 PPI, 95% porosity). Pressure drop data could be correlated with the superficial velocity by means of the Forchheimer equation. Empirical equations for the calculation of the viscous and inertial permeability as a function of the morphological parameters were derived. The pressure drop was very sensitive to imperfections within the foam packing, leading to considerable deviations of up to ±20% for foam samples having apparently the same morphology. The analogy between the mass and momentum transfer within foams was successfully evaluated with the Lévêque equation. Hence, mass transfer coefficients of foams can be estimated from pressure drop data, the latter being available with much less experimental effort. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction Open-cell ceramic or metallic foams have gained interest in the last decade due to their potential applications as carriers in various processes. Their properties such as low pressure drop, high surface area enhancing mass transport, as well as the radial mixing in the tortuous structure, improving heat transfer in highly endothermic and exothermic reactions, make them attractive in many research fields. They have already been used in several applications such as waste and exhaust gas purification (Pestryakov et al., 1996; Fino et al., 2005; van Setten et al., 2003), methane and propane combustion (Cerri et al., 2000; Schlegel et al., 1994), carbon dioxide reforming (Richardson et al., 2003a), partial oxidation of hydrocarbons at short contact times (Williams and Schmidt, 2006; Panuccio et al., 2006), preferential oxidation (PROX) of carbon monoxide in ∗ Corresponding author. Present address: BASF SE, 67056 Ludwigshafen, Germany. Tel.: +49 621 6079096. E-mail addresses: [email protected], [email protected] (F.C. Patcas).

0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.06.015

hydrogen (Sirijaruphan et al., 2005; Jhalani and Schmidt, 2005; ¨ Worner et al., 2003; Chin et al., 2006) and Fischer–Tropsch-synthesis (Chin et al., 2005). Most of these applications involve high flow rates and/or high reaction rates, typically controlled by external mass and heat transfer. The knowledge of mass (and heat) transfer and pressure drop properties of these reticulated structures is therefore of extreme importance for reactor design and industrial implementation. While the pressure drop in foams has been widely studied in the literature (Fourie and Du Plessis, 2002; Richardson et al., 2000; Decker et al., 2002; Schlegel et al., 1993; Giani et al., 2005; Moreira et al., 2004; Lacroix et al., 2007), mass transfer studies are rather scarce and incomplete. In previous works, Richardson et al. (2003b) reported mass and heat transfer properties of one ceramic foam with a cell density of 30 pores per inch (PPI). Tronconi and co-workers also studied mass transfer in three metallic and one ceramic foam structure (8–15 PPI), proposing a first correlation for the prediction of mass transfer in foams (Groppi et al., 2007). The convective and radiative heat transfer characteristics of foams with a great variety of pore densities and materials have been reported by different authors, mostly on the basis of dimensionless

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(Nusselt and Reynolds) numbers (Younis and Viskanta, 1993; Schlegel et al., 1993; Lu et al., 1998; Decker et al., 2002; Peng and Richardson, 2004). When the radiation influence is negligible, mass and heat transfer data can be compared by applying the Colburn analogy: Nu Re · Pr1/3

=

Sh Re · Sc1/3

.

(1)

Previous investigations from our group have shown that mass transfer coefficients and pressure drop of ceramic foams are between those of honeycomb monoliths and packed beds with comparable geometric surfaces (Patcas et al., 2007). Nevertheless, the mass transfer properties of these structures (Sherwood numbers) cannot be correlated to the hydrodynamic behavior of the system with Reynolds and Schmidt numbers as the only influencing parameters. The additional influence of the foam geometry has to be considered. The focus of this work is therefore to perform a quantitative characterization of the geometry (characteristic length, geometric surface area and porosity), the mass transfer and the pressure drop properties of foams with pore densities between 10 and 45 PPI and porosities between 75% and 85%. With the analyzed geometric parameters, we evaluate the influence of different geometric features on the mass transfer coefficients and the pressure drop in a wide range of Reynolds numbers and compare our results with mass (and heat) transfer and pressure drop data from the literature. The influence of the foam geometry is considered in the dimensionless Sh–Re evaluation with the aim to yield a correlation with broad applicability for the prediction of mass transfer coefficients in foams from hydrodynamic or pressure drop data. 2. Experimental 2.1. The foams The open-cell ceramic foams used in this work were fabricated by Vesuvius Inc. The foam monoliths were made of -Al2 O3 (99.5 wt%) as cylindrical pieces of 15 or 25 mm, respectively, in diameter and 50 mm in length. The cell densities and porosities (given by the manufacturer) of the analyzed foams were 10-, 20-, 30-, 45-PPI and 0.75, 0.8 and 0.85, respectively, as displayed in Fig. 1. For means of material comparison, a 40 PPI metal foam (stainless steel) provided by Glatt GmbH, with a porosity of 94.5% was also analyzed. 2.2. Characterization of the foam geometry 2.2.1. Characterization of the pore diameter by image analysis From different pieces of each foam type, 4–6 mm thin slices were cut and analyzed with a LEICA-DM-4000-M microscope. Images were taken from the slices using a LEICA DFC 280 digital camera with an optical magnification of 25. Only clearly focused pores and struts were analyzed from these images by using the Application Suite software from LEICA Microsystems Cambridge Ltd. Two orthogonal lengths were taken for each window or pore (pore is defined here as the communicating window between cells). The strut diameter was taken from the middle of the strut, being mostly the thinnest part between the vortices. Over 100 pores and struts were analyzed and statistically evaluated for each foam type in order to ensure a representative value. 2.2.2. Characterization of the geometrical surface area Sgeo with MRI The geometric surface area per bulk volume of ceramic foams is an important geometric feature of these structures and experimentally difficult to measure. It is the total external surface of the struts as if they were perfectly smooth. It cannot be determined by conventional methods like the nitrogen adsorption isotherms (BET) because

Fig. 1. 10-, 20- 30- and 45-PPI foams with porosities between 75% and 85% and 40-PPI metal foam (from the right to the left).

in reality the ceramic material is rough and the struts possess additional internal pores and channels stemming from the burnt polymeric template. Therefore, a volume imaging method is necessary. A technique commonly used for three-dimensional imaging is the X-ray absorption tomography and has already been performed on foams (Vicente et al., 2006). The magnetic resonance imaging (MRI) is another method already successfully used in the field of engineering on foams, and allows a three-dimensional imaging of the ceramic structure with acceptable image resolution (Groe et al., 2008). Dimensional image measurements were performed on every foam type with a Bruker Avance 200 SWB tomograph (magnetic flux density 4.7 T, micro-2.5 gradient system generating up to 1 T/m). A concise description of principles and measurement methods of MRI is given by Hardy (2006). Since the ceramic foams produce practically no signal in MRI experiments, the pore space was imaged by filling it with a liquid active for 1 H-MRI measurements. A bubblefree filling with degassed water under vacuum was performed on every cylindrical foam sample and copper sulphate (1 g/L) was added to the water allowing faster measurements due to enhanced relaxation. The resolution obtained was 50–86 m per voxel (threedimensional pixel), depending on the sample. The last step needed was the filling of the hollow struts. For this purpose, an algorithm was developed, using the fact that the voids in the struts were narrow and concave. The number of solid voxels (no MRI signal) in the cubic neighborhood (26 neighbors for each voxel) of each voidvoxel was counted. If there were more than 14 solid neighbors, the voxel was identified as void inside the struts and set to solid. The specific surface area of the filled foam structure matrix was determined using a Crofton formula, counting and averaging void–solid interfaces along various directions. A detailed description of the theoretical background, the used algorithms, and the principle ¨ of this method can be found elsewhere (Ohser and Mucklich, 2000). 2.2.3. Characterization of porosities with mercury porosimetry Since the ceramic foams were manufactured by using the replication method, the former polyurethane foam structure was decomposed after calcining leaving an additional void fraction in the solid strut network. Therefore, the ceramic supports used herein have a total porosity composed of the exterior, hydrodynamically relevant void space between the struts and an internal porosity of the strut ceramic. Using He-pycnometry, only the total porosity can be measured, since He reaches every void volume in the analyzed foam. Mercury porosimetry is a suitable method for measuring separately both porosities. Using a non-wetting fluid like mercury the external, hydrodynamically relevant porosity can be first obtained by measur-

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Log Differential Intrusion vs Diameter Log Differential Intrusion Cumulative Intrusion 0.065 0.10 0.060 0.09

0.055

0.08

0.050

0.040 0.06 0.035 0.05

0.030 0.025

0.04

Cumulative Intrusion (mL/g)

Log Differential Intrusion (mL/g)

0.045 0.07

0.020 0.03 0.015 0.02 0.010 0.01

0.005 0.000

0.00 1e+05

1e+04

1e+03

1e+02

Diameter (nm)

Fig. 2. A: Hg intrusion over the pore diameter of a 20PPI_0.8 sample. B: SEM image of the hollow triangular strut structure. C: SEM image of a pore between crystals of the -Al2 O3 ceramic.

ing the mercury intrusion in the megapores at atmospheric pressure. Since the hollow structure in the struts has diameters reaching from 50 to 200 m, higher pressures are needed to fill this void. At higher pressures, mercury fills every pore in the ceramic support, giving the internal strut porosity. The porosities were measured with a Micromeritics Autopore III 9420 porosimeter. The pressure range used was 0.007–1000 bar. Before mercury intrusion analysis, the cylindrical foam samples were weighted and the bulk volume was measured, giving the bulk density:

bulk =

mfoam 4 · mfoam = . 2 Vbulk  · Dsample · Lsample

(2)

After vacuum-degassing the penetrometer where the sample was placed, the system was filled with mercury. Since no pressure was applied, mercury only filled the open cells of the foam. The mercuryfilled sample was then removed from the apparatus and weighted. With known density of mercury and penetrometer volume, the density of the solid foam structure including the hollow void fraction within the struts was calculated to mfoam mfoam = . Vstrut Vpenetrom − VHg,1

where p is the imposed pressure,  is the surface tension (485 mN/m for Hg),  is the contact angle between Hg and Al2 O3 (130◦ ), and d is the diameter of the meso- and micropores. The decreasing diameters along the abscissa correspond to the increasing pressure during the filling process of the slightly porous solid structure of the foam. There are two pressure (or diameter) ranges with a steep increase in Hg intrusion. The first pores filled are of approx. 50–200 m. These pores represent the hollow strut network of the ceramic material, in accordance with microscope and SEM images from various samples (Fig. 2B). At pressures between 30 and 1000 bar, the pores of the not completely sintered ceramic material (12–600 nm) are filled (Fig. 2C). After intrusion, the density of the solid can be calculated by subtracting the total volume of mercury filled in low and the high pressure system (VHg,2 =VHg,1 +Vintrusion ) to the volume of the penetrometer,

solid =

mfoam mfoam = . Vsolid Vpenetrom − VHg,2

(5)

With these three density values for every foam type, the hydrodynamic and total porosities, h and t , respectively, can be determined to

(3)

h = 1 −

bulk , strut

(6)

The penetrometer was then placed in the high pressure system of the Autopore-porosimeter, where a controlled volume of Hg was pressed into the sample in small pressure steps. A typical intrusion plot is depicted in Fig. 2A. The imposed pressure is related to the pore size via the Washburn equation (Washburn, 1921):

t = 1 −

bulk . solid

(7)

strut =

p=−

4 ·  · cos  , d

(4)

2.3. Measurement of mass transfer properties of foams A suitable method for measuring external mass transfer properties from the bulk of the gas phase to the solid surface of a

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CO, air

blank foam

PFR

⋅ V0

⋅ Vin

CCO, 0

CCO, in

⋅ V0 CCO, out XCO

glass fabric

⋅ ⋅ VR = R ⋅ V0 Fig. 4. Molar balance of a PFR system with external recycle.

foam catalyst blank foam

air

thermocouples

Δp

Fig. 3. The packed reactor for kinetic tests. Fig. 5. Pressure drop device.

geometric structure is to quantify the effective rate of a chemical reaction catalyzed by the solid surface under conditions where mass transfer is the slowest and therefore rate limiting mechanism during reaction. The reaction chosen was the oxidation of carbon monoxide in air. The catalytic system Pt/SnO2 used as a washcoat to activate the foam surface is highly active for CO oxidation allowing mass transfer control at moderate temperatures. Preliminary kinetic experiments at different temperatures showed that at 220 ◦ C mass transfer control is reached for every structure at the highest Reynolds numbers. A detailed description of this procedure can be found in Patcas et al. (2007). 2.3.1. Preparation of the catalytic foams The foams were cut into thin cylindrical pieces of 15 or 25 mm in diameter from the middle part of the 50 mm samples supplied by the manufacturer. The pieces were then washed in distilled water and acetone, and subsequently calcined at 350 ◦ C to ensure a clean surface prior to the coating. The supports were then coated two times in a 20 wt% SnO2 -sol. After every coating, the pieces were dried at room temperature for 12 h and calcined at 350 ◦ C. Platinum was loaded on the SnO2 -coated precursors by adsorption of an organic Pt-complex from a supercritical solution in CO2 for 20 h, with subsequent in situ reduction of the organic rests with H2 . This process is known as supercritical fluid reactive deposition (SFRD) and is described in detail elsewhere (Patcas et al., 2007). 2.3.2. Reactor and process system The kinetic measurements of CO oxidation over Pt/SnO2 -coated foams were performed in a differential plug flow reactor with external recycling, operating as a CSTR system. The reactor was a glass tube of 16 (or 25) mm inner diameter and 250 mm in length. The tube was connected with two glass joints and stainless steel Swagelok fittings to the gas pipes. Fig. 3 shows a typically packed reactor. The thin catalyst cylinder (4–7 mm length) was carefully packed between the two blank foam neighbor pieces (20–30 mm) of the original foam cartridge to prevent discontinuity and open slits between the pieces, and then wrapped with thin glass fabric to avoid gas-bypassing dur-

ing measurements. After wrapping, the complete piece was pressed into the reactor. Foams and glass beads were placed above and below the packed segment of the tube for a better distribution and preheating of the gas before entering the catalytic segment. The reactor was then incorporated into the reaction unit. Thermocouples were laterally inserted into the reactor at the entrance and the outlet of the catalyst bed, and heating and isolation were placed around the reactor. The gas supply system comprised a mixture of CO in N2 and air, regulated with mass flow controllers from Brooks. Carbon monoxide was purified by a packed bed of quartz beads heated at 305 ◦ C to decompose iron carbonyls. The CO inlet concentration was varied between 1000 and 2000 vol.-ppm. The external recycle of the CO/air mixture was performed by a membrane pump. At a constant system gas hourly space velocity (GHSV) of approx. 300, 000 h−1 , the hydrodynamic conditions along the catalyst bed could be varied by altering the frequency of the pump and hence, the recycle flow rate (5–200 L/min (STP)), measured with M&W low-p mass flow meter. This allowed the variation of the superficial velocity in the reactor at the reaction temperature of 220 ◦ C. The inlet and outlet CO and CO2 concentrations were measured by a non-dispersive infrared spectrometer (NDIR) ABB-URAS-14. The kinetic coefficients of the CO oxidation in the mass transfer controlled regime were measured at superficial velocities between 0.2 and 12 m/s at reactor conditions (220 ◦ C). The volumetric rate coefficient based on the bulk volume of the foam catalyst can be calculated from the molar balance of the recycle system (Fig. 4) as kv =

(R + 1)V˙ 0 ln VBulk



CCO,in



CCO,out   1 + R(1 − XCO ) (R + 1)V˙ 0 = ln , VBulk (R + 1)(1 − XCO )

(8)

where R is the recycle ratio, Vbulk the catalyst bed volume, V˙ 0 the flow rate at reactor conditions and XCO the carbon monoxide conversion.

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2.4. Measurement of pressure drop properties of the foams The pressure drop was measured using the same apparatus. For this purpose, the upper and lower ends of a cylindrical foam piece of 50 mm were removed, since these showed increased blockage of pores due to the manufacturing process. The foam was then wrapped in glass fabric and pressed into a glass reactor with two lateral glass tubes placed 12 mm above and below the packed foam (Fig. 5). The assembling of the reactor was the same as described above. The pressure drop was measured with a water manometer at superficial velocities up to 15 m/s. The gas temperature was kept constant at 220 ◦ C in order to achieve the same hydrodynamic conditions as in the mass transfer experiments, for a better comparison of both properties.

3. Results and discussion 3.1. Geometric characterization of the analyzed foams 3.1.1. Pore and strut diameters Fig. 6 shows exemplarily the inner pore and strut diameter distributions of the 30 PPI foam with 80% porosity. The density of probability from a Gaussian normal distribution having the experimentally determined values for the mean diameter and the deviation is also plotted in the figure for comparison. While the struts have a rather narrow size distribution, the size of the windows varies within one order of magnitude (0.2–2 mm). This tendency was observed among all samples analyzed. The obtained values for this foam were calculated to dp =0.871 mm, with a root mean square deviation:  = 0.327 mm (37.6%). dt = 0.319 mm, with  = 0.075 mm (23.7%). The mean pore diameter, used as the characteristic length for the foams, is defined as the sum of mean inner pore and strut diameter and represents thus the average distance between the opposite struts delimitating a pore (Fig. 7):

3.1.2. Sgeo : MRI analysis and model comparison Fig. 8 shows a two-dimensional section of the three-dimensional MRI scan of the 20PPI_0.75 foam, as well as the same after processing the data with the algorithm used for the filling of the hollow struts. The measured geometric surfaces per bulk volume Sgeo,MRI of the Al2 O3 foams analyzed in this work are listed in Table 1. Until now, the geometric surface of foams has been estimated from theoretical models based on regular unit cells. A dense packing of tetrakaidecahedra (TTKD) is one of the geometries that best represent open-cell foams (Gibson and Ashby, 1988). From this model, an expression for the geometrical surface per bulk volume was developed leading to  bulk 4.82  4.82 · = · 1 − . (9) Sgeo,TTKD = Dp solid Dp A detailed description of the morphological assumptions made is given by Buciuman and Kraushaar-Czarnetzki (2003). A simpler model recently proposed for calculating the geometrical surface per unit volume of foams is the cubic cell (CC) model, originally developed for metallic foams by Lu et al. (1998) and later used by Giani et al. (2005) and Lacroix et al. (2007), giving Sgeo,CC =

4 · (1 − ). dt

The experimental Sgeo,MRI were compared to the geometrical models described above. Therefore, the experimental values of Dp were taken for Eq. (9). For calculating Sgeo,CC , both the measured strut diameters dt and the strut diameter defined by Lacroix et al. as a function of

dp

dt

Dp = dp + dt = 1.190 mm. The relative deviations of the pore size distributions of the foams varied between 36% and 47%. The main geometric features of all foams used in this work are listed in Table 1.

Fig. 7. Inner pore and strut diameters from image analysis. Both orthogonal pore diameters were considered in the statistical evaluation.

10% 9%

(10)

25% Inner pores

30 PPI relative frequency

30 PPI relative frequency

Struts

density of probability

density of probability

8%

20%

6%

frequency [-]

frequency [-]

7%

5% 4%

15%

10%

3% 2%

5%

1% 0% 0.259

0.613

0.966 1.319 Pore diameter dp [mm]

1.673

2.026

0% 0.037

0.180

0.322 0.464 Strut diameter dt [mm]

Fig. 6. Inner pore (left) and strut (right) size distributions of the 30PPI_0.8 foam.

0.606

0.748

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Table 1 Geometric and porosity features of the analyzed foams Foam typea

t b (–) (total)

h b (–) (hydrod.)

dp c (mm)

dt c (mm)

Dp d (mm)

Sgeo,MRI e (m−1 )

Sgeo,TTKD f (m−1 )

Al2 O3 10PPI_0.8 20PPI_0.8 30PPI_0.8 45PPI_0.8 10PPI_0.85 20PPI_0.85 30PPI_0.85 45PPI_0.85 20PPI_0.75

0.818 0.804 0.816 0.813 0.852 0.858 0.852 0.848 0.777

0.772 0.751 0.766 0.761 0.812 0.814 0.807 0.801 0.719

1.933 1.192 0.871 0.666 2.252 1.131 0.861 0.687 1.069

0.835 0.418 0.319 0.201 0.880 0.451 0.330 0.206 0.460

2.768 1.610 1.190 0.867 3.132 1.582 1.191 0.893 1.529

675.4 1187.0 1437.8 1884.3 629.3 1109.1 1422.4 1816.3 1290.3

831.5 1493.9 1959.3 2717.9 667.3 1314.0 1777.9 2407.8 1671.1

Stainless steel M_40PPI_0.95

0.946

0.917

0.628

0.174

0.802

a

–

1731.5

Pore count (PPI) and total porosity given by the manufacturer; “M” stands for metal.

b

From Hg-porosimetry.

c

From microscopy image analysis: > 100 pores and struts for each foam type.

d

Chosen as the characteristic length for dimensional analysis.

e

From MRI measurements.

f

Calculated with tetrakaidecahedron model using Dp and h .

Unfilled 20PPI-075 110 100 90 80 70 60 50 40 30 20 10 20

40

60

80 100 120 140 160 180 200 220 Filled 20PPI-075

110 100 90 80 70 60 50 40 30 20 10 20

40

60

80 100 120 140 160 180 200 220

Fig. 8. Example of data processing of the hollow strut structure from MRI analysis of 20PPI_0.75 foam.

inner pore diameter dp and porosity  (Eq. (11)) were tested, the latter resulting in surface areas closer to the experimental values for Sgeo from this work: 

4 · (1 − ) 3 dt,Lacroix = · dp . ·  4 1− · (1 − ) 3·

stituted by the density of the real struts of the ceramic foams used herein, which have a hollow strut network inside and therefore a lower density. The comparison between the measured Sgeo,MRI and the calculated surfaces Sgeo,TTKD and Sgeo,CC of foams with pore densities between 10 and 45 PPI and porosities between 75% and 85% is shown in Fig. 9A. There is an overestimation of the geometrical surface by both morphological models. One of the reasons for this overestimation is the assumption of struts with a constant thickness and a cross-section without accumulation of solid. Ceramic foams commonly have considerable accumulation of solid material in the area of the vortices, where the struts are connected. On the other hand, some pores in real foams are closed, since the ceramic slip can form bubbles in the cells during coating of the polymer matrix. Lacroix et al. (2007) also observed the non-homogeneous deposition of matter upon re-infiltration of his SiC foams, and mentioned this as the reason for the considerable deviations between their calculated and measured strut diameters at lower porosities. These facts lead to a lower strut surface in real foams as compared to an ideal network of cubes or tetrakaidecahedra. Not only are the absolute values of the surface overestimated, but also the influence of the void fraction. Although the measured Sgeo increase slightly with the porosity, the dependence is not as strong as predicted by the models. It is worth noting that the CC model results in a stronger overprediction of the surface area than the tetrakaidecahedron model, as clearly recognizable in Fig. 9A. Since the tetrakaidecahedron model displays the most efficient space filling regular network of unit cells, having the lowest geometric surface per volume (Gibson and Ashby, 1988), it is still most probably the best idealized representation of the foam geometry. The measured dependency on the pore diameter is somewhat lower than 1/Dp . An empirical model between the geometrical surfaces Sgeo,MRI and the measured pore diameters Dp and external porosities h was developed by a simple parameter fitting to the experimental values, resulting in a good correlation of the geometric features of the foams (Fig. 9B).  Sgeo = 3.84 ·

(11)

Since the models assume a solid strut structure (no hollow volume inside the strut), the solid density in Eqs. (9)–(11) has to be sub-

 Dp −0.85 −0.82 · h . m

(12)

The metal foam (stainless steel) M_40PPI_0.95 could not be measured with MRI due to the nature of its material. Therefore, the geometrical surface of this foam was calculated from both models (empirical and tetrakaidecahedron) with experimental values for Dp and h . The difference between the calculated surfaces is negligible.

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3500

Al2O3 20PPI_0.75 Al2O3 10-45PPI_0.8 Al2O3 10-45PPI_0.85 0.75 TTKD model 0.80 TTKD model 0.85 TTKD model 0.75 CC model 0.80 CC model 0.85 CC model

3000

Sgeo [m-1]

2500 2000 1500

2500

Al2O3_10-45PPI_0.8 Al2O3_10-45PPI_0.85 Al2O3_20PPI_0.75 M_40PPI_0.95 empirical model M_40PPI_0.95 tetrakaid. model Empirical model

2000 Sgeo [m-1]

5208

1500 1000

1000 500 500

Sgeo = 3.84·(Dp/m)

-0.85

-0.82

·εh

0

0 0

0.001

0.002 Dp [m]

0.003

0

0.004

200

400

600

(Dp/m)-0.85·ε-0.89

Fig. 9. A: Comparison between Sgeo,MRI (black symbols), Sgeo,TTKD (hollow symbols) and Sgeo,CC (gray symbols) for all analyzed foams. B: Correlation between measured values for Sgeo,MRI , Dp and h of ceramic foams.

Table 2 Bulk density and porosity variations of open cell ceramic foams Foam

Axial position

bulk (g/cm3 )

strut (g/cm3 )

solid (g/cm3 )

h (–) (hydrod.)

t (–) (total)

20PPI_0.8 20PPI_0.8 20PPI_0.8 20PPI_0.8 20PPI_0.85 20PPI_0.85 45PPI_0.85 45PPI_0.85

End End Middle Middle End Middle End Middle

0.780 0.870 0.733 0.740 0.607 0.538 0.710 0.447

3.081 3.177 3.066 3.111 3.055 3.010 3.053 2.941

3.934 3.976 3.945 3.970 3.933 3.925 3.989 3.976

0.747 0.726 0.761 0.762 0.801 0.821 0.767 0.848

0.802 0.781 0.814 0.814 0.846 0.863 0.822 0.888

Since the porosity of metal foams is high (usually > 90%) in comparison to ceramic foams, the struts have a lower concavity and accumulation of solid at the vortices, leading to a strut structure closer to the ideal morphology of a tetrakaidecahedron. This is most probably the reason for the good agreement between both models, when applied to the metallic foam.

3.1.3. Porosity via mercury porosimetry The value for the solid density remains practically unchanged since the ceramic material used is the same. The strut density is also similar for all foams, with a maximum variation of 5.4% (Table 2). While the average porosities of the 50 mm cylindrical foams as supplied by the manufacturer had the total porosity specified, significant deviations from these porosities were observed along different axial locations of the cylindrical piece after cutting. The variations arise from the anisotropic bulk density along the foam that results during removal of excess ceramic slurry in the manufacturing process, leading to higher densities (lower porosities) in the edges of the cylinders and lower densities in the middle part. Between a 15 mm piece from the edge and one from the middle of a 50 mm foam as delivered, the maximum bulk density variation observed was as high as 45%, leading to an absolute porosity variation of 10%. The inhomogeneity of the cylindrical foams usually increased with increasing pore count. Typical results of these variations are shown in Table 2. For this reason, the evaluation of the bulk densities of 6–8 different foam pieces per foam type was necessary in order to get representative averages for the real void fractions of these structures. The mean values for the external and total porosities of the analyzed foams are given in Table 1.

3.2. Mass transfer in ceramic foams 3.2.1. Washcoating of the foam catalysts The SnO2 content of the used foam samples after two loadings varied between 3.5 (10 PPI_0.85) and 8.8 wt% (45PPI_0.8), increasing with the geometric surface of the foams, as would be expected in the case of a SnO2 layer of constant thickness. The layer thickness was observed on different samples with SEM, being constant for all catalysts. Along the strut network the thickness of the coat was approx. 5–7 m, where at regions with higher concavity the value could increase up to 15 m (Patcas et al., 2007). The Pt content of the SnO2 -coated precursor foams after supercritical deposition and reduction varied between 0.4 (10 PPI_0.85) and 1.1 wt% (45PPI_0.8), with a nearly constant ratio Pt/SnO2 of 10–13%. Image analysis of the foams before and after washcoating showed negligible variation (< 2%) of the strut and inner pore diameters. The mean pore diameter Dp showed obviously no variation.

3.2.2. Mass transfer coefficients Fig. 10 displays the volume based mass transfer coefficients of the foams with pore densities between 10 and 45 PPI and a total porosity of 85% as a function of the superficial gas velocity. Different shadings within a point symbol denominate different catalytic pieces of the same type. Due to the irregular nature of these structures, deviations between different samples were expected. The deviations are small but increase with increasing pore count from ±3% (10 PPI) to ±6–7% (45 PPI), in parallel with the inhomogeneities of the bulk density observed on the foam bodies provided by the manufacturer. For this reason, three to four different foam samples of every foam type were measured in order to gain representative values for their mass transfer properties.

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

1000

5209

0.6 45PPI_0.85 30PPI_0.85

km,30 = 0.2177*U00.3577

0.5

20PPI_0.85

800

10PPI_0.85

0.4 kv [1/s]

km [m/s]

600

km,45 = 0.2062*U00.3887

0.3

400

km,20 = 0.1749*U00.4678

0.2

45PPI_0.8

200

30PPI_0.8

0.1

20PPI_0.8 km,10 = 0.163*U00.5062

10PPI_0.8

0.0

0 0

4

8

0

12

4

8

U0 [m/s] Fig. 10. Volumetric mass transfer coefficients kv of foams versus superficial gas velocity.

12

U0 [m/s] Fig. 11. Mass transfer coefficients km of foams with 80% porosity versus superficial gas velocity.

0.6 As expected from the external mass transfer control, the rate coefficients increase with increasing superficial velocity: kv ∼ (0.4−0.52)

kv . km = Sgeo

0.4 km [m/s]

U0 . The volume based coefficients increase with decreasing pore size, since the geometrical surface area per bulk volume increases as well. A higher amount of CO can be converted in the same volume of a 45 PPI foam than in a 10 PPI foam. The foams with lower porosity showed the same tendency and very similar values to the foams with 85% porosity. With the measured geometrical surfaces Sgeo from MRI analysis, the mass transfer coefficients could be calculated as

km,0.85 = 0.1666*U00.4431

0.5

0.3 km,0.8 = 0.151*U00.4758 0.2

(13)

km,0.75 = 0.1458*U00.4689

0.1 The ceramic foams exhibit similar mass transfer coefficients km over the whole velocity range even though the surface areas differ considerably, as is exemplarily shown in Fig. 11. The dependence of the mass transfer coefficients on the superficial velocity is slightly higher for foams with bigger pores (10 and 20 PPI) than for the other two pore densities. The same tendency was observed by Schlegel et al. (1993) when measuring heat transfer coefficients on cordierite foams with pore counts very similar to those used in our work. The authors found similar heat transfer coefficients for all foams in a range of 0.5–14 m/s, as well as increasing U0 -exponents with increasing pore size. The influence of the porosity is weak. The mass transfer coefficients of the 20 PPI foams increase slightly with the porosity at a constant interstitial velocity. Fig. 12 depicts the average km slopes of 20 PPI foams with three different porosities versus the interstitial velocity U0 / h . Excluding the dependence of the mass transfer properties of foams on reactor diameter and length of catalyst bed is essential for the applicability of the results given in this work. For this reason, different foam sample diameters and lengths were chosen for the foams of types 20 PPI_0.8 and 30PPI_0.8. Two reactor diameters as well as three bed lengths were chosen: 15 and 25 mm, and 14, 25 and 50 mm, respectively. The influence of both parameters is negligible as can be seen in Fig. 13. The volume based mass transfer coefficients show similar values. The deviations are smaller than 10%, within the deviations found upon measurements of different foam samples of the same type. Therefore, it can be concluded that, within

20PPI 0 0

4

8 U0/ε [m/s]

12

16

Fig. 12. Mass transfer coefficients km of 20 PPI foams with 75%, 80% and 85% porosity over the interstitial gas velocity.

the evaluated range, the characteristic flow of the analyzed foams is fully developed, and the mass transfer properties are not dependent on the reactor geometry. 3.2.3. Dimensional analysis: Sh–Re evaluation of experimental results The mass transfer properties of a given structure depend on the fluid properties (density, viscosity, diffusion coefficient) of the flowing medium, the superficial velocity and the geometric features of the structure. A dimensional analysis can help to establish a quantitative relationship between the mass transfer properties of geometrically similar structures, and the related dimensionless products of the variables, thereby reducing the number of independent parameters. The dimensionless form of the mass transfer coefficient is the Sherwood number Sh =

km · Dp D1,2

(14)

5210

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217 Table 3 Values for A and B from Eq. (15) for the analyzed foams

1000 20PPI_0.8_D = 15mm_L = 14mm 20PPI_0.8_D = 15mm_L = 50mm 20PPI_0.8_D = 25mm_L = 50mm

800

kv [1/s]

600

Foam type

A

B

Re-range

10PPI_0.85 20PPI_0.85 30PPI_0.85 45PPI_0.85 10PPI_0.8 20PPI_0.8 30PPI_0.8 45PPI_0.8 20PPI_0.75

1.60 ± 0.16 1.26 ± 0.16 1.15 ± 0.14 1.04 ± 0.11 1.34 ± 0.09 1.65 ± 0.14 1.21 ± 0.18 0.97 ± 0.13 1.17 ± 0.22

0.48 ± 0.02 0.45 ± 0.02 0.45 ± 0.03 0.43 ± 0.02 0.50 ± 0.01 0.41 ± 0.01 0.43 ± 0.03 0.44 ± 0.03 0.46 ± 0.03

77–1103 62–661 7.5–395 8.9–325 66–946 58–695 54–507 36–352 68–620

400

200

0 0

4

8

12

U0 [m/s] Fig. 13. Mass transfer coefficients of 20 PPI_0.8 foams with different dimensions.

Sh [-]

100

10PPI_0.85 20PPI_0.85 30PPI_0.85 45PPI_0.85 10PPI_0.8 20PPI_0.8 30PPI_0.8 45PPI_0.8 20PPI_0.75

10

1 1

10

100 Re [-]

1000

10000

Fig. 14. Experimental Sh numbers as a function of the Re numbers of nine different foam types: circles—10 PPI, diamonds—20 PPI, triangles—30 PPI and squares—45 PPI. Different shadings represent different porosities.

and can be expressed as a function of the dimensionless hydrodynamic properties, or Reynolds number (Eq. (16)) and the fluid properties, or Schmidt number (Eq. (17)) as Sh = A · ReB · Sc1/3 Re = Sc =

U0 · Dp



D1,2

.

and

with

(15) (16) (17)

The diffusion coefficient of CO in air at system gas temperature is 4.1–4.5×10−5 m2 /s, depending on the system variations in pressure. By using the experimentally found values for km of the foams with different porosities and pore sizes, their Sh numbers were plotted over the experimental Re numbers. Since the minimum superficial velocity in the operation modus with external recycle was 1.2 m/s, two foams (30PPI_0.85 and 45PPI_0.85) were tested in a plug-flow system without recycling, to extend the velocity range towards lower values. The conversion data in this case were transformed into kinetic

rate constants by using the PFR reactor model. The minimum superficial velocity was 0.2 m/s, since at lower velocities (higher residence time) the CO conversions were higher than 95%. Fig. 14 shows exemplarily one set of experimental Sh–Re data for each foam type. The Sherwood numbers of the foams increase with the size of the pores. The values for A and B (Eq. (15)) found for each foam type (three data sets for every foam type) are listed in Table 3. In comparison to spherical particle packings, who's Shp numbers can be described with Rep and Sc as the only influencing parameters, the mass transfer properties of foams cannot be correlated with the hydrodynamic(Re) and fluid- (Sc) properties alone. 3.2.4. Comparison with literature results When comparing with literature data for mass or heat transfer, the geometrical and material properties of the structures from the literature have to be known as precisely as possible. The geometric specifications of the authors are often insufficient, and models for calculating the geometric surface of foams differ considerably, making a comparison difficult. For this reason, before comparing the data from this work with other publications, the information given by the authors is considered and listed in Table 4. Groppi et al. (2007) proposed a generalized correlation for mass transfer in metallic and ceramic foams. Four different foams were taken there into account: three metallic foams (Fe–Cr-alloy) from a previous publication (Giani et al., 2005), and one ceramic foam (-Al2 O3 ) with a pore count of 8.2 PPI (Table 4). The correlation proposed used the strut diameter as the characteristic length and the maximum interstitial velocity between struts for Re definition: 1/3 . Sh = 0.91 · Re0.43 max · Sc

(18)

To compare these data with ours, the correlation had to be redefined with the pore diameter of the CC and the superficial velocity in Sh and Re. The correlation of Groppi is in good agreement with the mean Sh values of the 10 PPI foams from this work (Fig. 15), since those authors used mainly foams with large pore sizes (8–15 PPI). Although the geometric features of the system, such as the foam diameter and length (Table 4), as well as the procedure of reactor loading differed considerably, the obtained values for Sh match satisfactorily. Schlegel et al. (1993) analyzed the convective heat transfer and the pressure drop in ceramic foams made of cordierite having similar pore densities and porosities as the foams analyzed in this work. Since the heat transfer measurements were performed at 160 ◦ C, radiation is negligible, so the analogy between heat and mass transfer, or Colburn analogy (Chilton and Colburn, 1934), can be applied for data comparison. Despite of completely different experimental methods and ceramic materials, the authors also observed an increase in heat transfer (Nu) with decreasing pore count, as can be seen in Fig. 15. The Nu values of these authors are qualitatively in good agreement with the experimental Sh numbers from this work. The slope of the10 PPI foam is similar in all cases, even though the geometry of the sample varies extremely from 9 (Groppi) to 76 mm (Schlegel) in diameter. The deviations at higher pore densities

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

5211

Table 4 Geometric properties of foams analyzed in the quoted literature Mat.

Porosity t (–)

(PPI)

Dp,h a (mm)

Dstrut (mm)

Sgeo (m−1 )

Re-exponent

(DF /LF )b (mm/mm)

Groppi et al. (2007) Mass transfer (Sh) 8.2 Al2 O3

0.84

3.1

0.84

802c

0.43

9/6–25

FeCr FeCr FeCr

0.95 0.94 0.93

4.3 2.0 1.7

0.73 0.29 0.23

336c 761c 962c

0.43 0.43 0.43

9/6–25 9/6–25 9/6–25

Schlegel et al. (1993) Heat transfer (Nu) Cord. 10

0.858

5.2

–

667d

0.47

76/150

Cord. Cord. Cord.

0.867 0.863 0.854

2.6 1.73 1.02

– – –

1334d 2001d 3335d

0.43 0.42 0.27

76/165 76/153 76/150

a

5.9 12.5 15.3

20 30 50

Dp

Dh

p: pore; h: hydraulic.

b

Refers to the diameter of the foam piece divided by its length.

c

Calculated from the cubic cell model.

d

Manufacturer specifications (Selee) idem.

Sh, Nu [-]

100 10PPI_0.85 20PPI_0.85 30PPI_0.85 45PPI_0.85 10PPI_0.8 20PPI_0.8 30PPI_0.8 45PPI_0.8 20PPI_0.75

10

Groppi: 8.2 PPI Schl.: 10 PPI Schl.: 20 PPI Schl.: 30 PPI Schl.: 50 PPI

1 1

10

100 Re [-]

1000

10000

Fig. 15. Comparison between mass and heat transfer data: this work (Sh), Groppi (Sh) and Schlegel (Nu).

transfer properties of the foams: dt , dp , Dp and (Sgeo )−1 . The first one tested was the strut diameter, based on the good correlation achieved with it in Groppi et al. (2007). With the foams analyzed herein, no general description was possible with the strut diameter as the characteristic length in Sh and Re. Due to the nearly constant ratio between all these “characteristic” lengths, the use of any of them yielded the same tendency, namely different Sh–Re slopes for different foam types, as shown in Figs. 14 and 15. The pore diameter perpendicular to the direction of flow was chosen as Lchar , since it is experimentally accessible and represents in our opinion a combination of two characteristic geometric features for the void and solid fractions in foams (dp + dt ). In order to describe these geometrically not perfectly similar cellular materials in a more general way, an additional geometric influencing factor, defined here as Fg , has to be included in Eq. (15): Sh = A · ReB · Sc1/3 · Fg .

(19)

From the analyzed geometric features of the foams (Table 1), various definitions for Fg can be chosen and evaluated by parameter fitting. between the Nu numbers reported by Schlegel and the Sh numbers measured herein cannot be overlooked. The differences are most probably due to the different geometrical surfaces used. The authors did not specify the origin of the values for Sgeo , but at a constant pore count, their calculated surfaces are considerably higher than the ones measured herein, as can be seen upon comparison between Tables 1 and 4. The higher Sgeo values result in lower heat transfer coefficients and thus, lower Nu numbers. Furthermore, the authors did not measure the actual pore diameter Dp . They defined Nu and Re with the hydraulic diameter Dh = 4 ∗ /Sgeo , which hinders the quantitative comparison between Sh numbers from this work and their experimental Nu numbers. From the discussion in thissection it becomes clear that the mass transfer properties of foams cannot be correlated with Re and Sc as only parameters. In an attempt to find a correlation describing all foams consistently, the additional geometric influence has to be studied. 3.2.5. Correlation of the mass transfer properties of ceramic foams If the foams would be geometric perfectly similar structures, i.e. every dimension of the foams would enlarge or contract by exactly the same factor, the dimensionless representation of mass transfer in foams would be generally described with Sh = f (Re, Sc). All characterized geometric features were tested for correlating the mass

3.2.5.1. Mathematical modeling. The fitting of the model parameters was done by using the Matlab software. A large number of different geometric functions Fg were defined, containing different combinations of the four geometric features mentioned above and the parameters to be fitted (m, n), used as powers. The parameters that were fitted were: A and B from Eq. (15) and the parameters from the Fg definition. The starting values for the parameters fitting were varied from zero to unity, showing no changes in the end values. The nonlinear minimization was done with the function “lsqnonlin”. The algorithm in this function is a trust region method based on the interior-reflective Newton method, described in Coleman and Li (1996). With this algorithm, the confidence interval limits as well as the correlation of the estimated parameters, using the Jacobian matrix, can be calculated. Introducing Fg simply as the ratio of the characterized lengths resulted merely in a further constant, as described above, with insignificant influence and therefore not unifying mass transfer data. The optimum Fg giving the lowest absolute variance (0.76) was found to be  Fg =

 Dp [m] m · (h )n . 0.001m

(20)

5212

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

Table 5 Estimated parameters for the correlation of ceramic foams

100

Value

Confidence interval (95%)

A B m n

1.00 0.47 0.58 0.44

±0.05 ±0.01 ±0.01 ±0.11

Shfoams/ (Sc1/3*Fg) [-]

Parameter

Table 6 Symmetric correlation matrix C of the four estimated parameters A A B m n

1 – – –

B −0.770 1 – –

m 0.020 −0.489 1 –

Correlation CI Upper L CI Lower L 45PPI_0.85 30PPI_0.85 20PPI_0.85 10PPI_0.85 45PPI_0.8 30PPI_0.8 20PPI_0.8 10PPI_0.8 20PPI_0.75

10

Sh

n

1/3

Sc

0.643 −0.035 −0.385 1

0.44

⋅ εh

⋅ (Dp/0.001m)

0.58

= 1.00⋅Re

0.468

1 1

10

100 Re [-]

1000

10000

Fig. 16. Correlation for the prediction of mass transfer coefficients in ceramic foams.

3.2.5.2. Mass transfer correlation for ceramic foams. Pore and cell anisotropy and variation of cell orientation of different foam geometries were optically recognized. Nevertheless, an additional and extremely laborious (and representative) characterization of pore diameters in the x–z and y–z plane would be needed for every foam type in order to quantify the anisotropy and preferential orientation of the pores in space. With this, influencing dimensionless geometric ratio(s) for this phenomenon could probably be found out and included, to describe these geometrically not perfectly similar cellular materials in a more general way. Such a three-dimensional characterization of the pores of 10 different foam geometries is not within the scope of this research. The attempt of this work is to find a correlation for the prediction of mass transfer properties in foams with different geometric features, which the user can take advantage of in the simplest way possible. The introduction of (Dp /0.001m) is not physically founded, but unifies the mass transfer data collected in this work, yielding a correlation with a minimum of geometric parameters needed, most advantageous for the user, who would otherwise need to do a complex geometric characterization of the used structure before using the correlation to predict mass or heat transfer coefficients. The proposed correlation holds for all ceramic foams irrespective of their porosities or pore counts. The mass transfer properties of

100 Correlation CI Upper L

Shfoams/ (Sc1/3*Fg) [-]

The estimated parameters and their confidence intervals are listed in Table 5. The parameter errors were estimated by a 95% confidence limit. It is important to know how the parameters are correlated to each other. This can be quantified by the variance–covariance matrix V, obtained from the Jacobian matrix J (N × 4) and the experimental error variance Var as V = Var · (JT · J)−1 . This 4 × 4 matrix allows the calculation of the relative correlation of the parameters, by dividing every element of the matrix by the square root of the product of  the variances of the parameters: Cij = covij / vari · varj , i and j being the indices of different parameters. Table 6 displays the relative correlation matrix C of the estimated parameters to each other. The diagonal elements are one, since the covariance of one parameter to itself is equal to its variance. If an element of the matrix (i, j) is close to unity, the parameters i and j are strongly correlated. If the element is close to zero, the correlation of the parameters to each other is weak. Parameters A and B are strongly correlated, since for an optimum representation of the data an increase in A would result in a decrease in B in order to keep the deviation between the correlation in the data points as low as possible. The same logical relation applies for the correlation between the remaining parameters, whereas the influence of their correlation is not determining for the quality of the data representation.

CI Lower L M_40PPI_0.95

10

1 10

100 Re [-]

1000

Fig. 17. Comparison between experimental Sh values of metal foam M_40PPI_0.95 and the proposed correlation.

open-celled ceramic foams with pore sizes Dp between 0.87 and 3.13 mm (10–45 PPI) and total porosities between 75% and 85% can be predicted by the relation: Shfoam = 1.00 · Re0.47 · Sc1/3 ·



0.58 Dp · 0.44 h 0.001m

(21)

in the Reynolds range: 7 < Re < 1100. The slope of the correlation together with the data for all foams is depicted in Fig. 16. In an attempt to verify whether the correlation for ceramic foams can be also used to predict mass transfer coefficients of metallic foams, the mass transfer properties of the stainless steel foam M_40PPI_0.95 were also analyzed. The geometrical surface based on the tetrakaidecahedron model was used to calculate the experimental km values of the metal foam from the data for kv , since the empirical model developed from the MRI characterization was done for ceramic foams only, and the deviation between the empirical and the tetrakaidecahedron model was merely 2.4% when applied to the metal foam (Fig. 9B). The experimental Sh data of the metal foam were taken and plotted for comparison together with the established correlation (no data fitting) in Fig. 17. The agreement between the slopes is within 6%. As can be seen, there is no systematic error and the experimental points fall pretty well within the confidence interval of 95%. The correlation (Eq. (21)) established for ceramic foams can be used for the estimation of mass transport properties of metal foams as well.

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

14

18 30PPI_0.8 (697 kg/m3)

12

20PPI_0.8

14

30PPI_0.8 (707 kg/m3)

10

20PPI_0.75

16

30PPI_0.8 (718 kg/m3)

Δp/ΔL [Pa/m]*10-4

Δp/ΔL [Pa/m]*10-4

5213

3

30PPI_0.8 (714 kg/m )

8 6 4

20PPI_0.85

12 10 8 6 4

2 2

Δp/ΔL = 3719.4*U0 + 545.9*U02

0 0

2

4

6 U0 [m/s]

8

10

12

0 0

6

9

12

15

U0 [m/s]

Fig. 18. Pressure drop versus superficial velocity of the 30PPI_0.8 foam (220 ◦ C).

Fig. 20. Influence of porosity on pressure drop.

The pressure drop of a fluid when flowing through a porous structure can be quantified with the Forchheimer equation:

18 εt = 0.8

16

p p

 = f · U + f · U 2 or = ·U + · U2 , L 1 0 2 0  L k1 0 k2 0

45 PPI 30 PPI 20 PPI 10 PPI

14 Δp/ΔL [Pa/m]*10-4

3

12

where p is the pressure drop, L the measured length of the structure, U0 the superficial velocity and f1 and f2 the constants for one foam type at constant hydrodynamic conditions. The mean values for the constants from all samples measured were taken for the description of the average pressure drop of a given foam type. For the 30PPI_0.8 foam, f1 and f2 are given in Fig. 18.

10 8 6 4 2 0 0

(22)

3

6

9

12

15

U0 [m/s] Fig. 19. Pressure drop of 10–45 PPI foams with 80% porosity.

3.3. Pressure drop in ceramic foams The pressure drop measurements were reproducible on the same foam sample within 1%. The influence of the 3–8 wt% SnO2 washcoat on the pressure drop of a foam sample was evaluated and is negligible (< 4%). Due to variations between different samples of the same type, 3–5 different foam samples had to be measured of every foam type. Fig. 18 shows the data of 30PPI_0.8 (four different samples). Since the deviations upon all measured foams were up to ±15%, the bulk density of every analyzed foam piece was evaluated in order to find out if the deviations were connected to variations in the bulk density of the pieces. As clearly displayed in the figure, the deviations cannot be correlated to variations in density. The reason for the discrepancy is most probably the irregular structure of the pores. For instance, some pores in the foam pieces are closed, leading to a higher drag coefficient cD . The number of irregularities, or closed pores, is not constant over different samples and has a much stronger effect on the pressure drop than it has on the mass transfer properties. Other authors have reported similar observations upon their pressure drop data (Richardson et al., 2000). For this reason, the pressure drop behavior of a foam cartridge is a good tool to check the foam on the presence of closed pores inside.

3.3.1. The influence of the pore count Typical pressure drop values of the foams with 80% porosity are shown in Fig. 19. As expected, the pressure drop increases as the pore size decreases, since the geometric surface increases with the pore count. Nevertheless it is important to note that the 30 and 45 PPI have similar pressure drop slopes. In singular cases, the pressure drop over a 30 PPI sample was even higher than over a 45 PPI foam. 3.3.2. The influence of porosity Fig. 20 displays exemplarily the influence of porosity on the pressure drop of 20 PPI foams. The pressure drop evidently increases with decreasing void fraction. The deviation between different samples also increases strongly at lower porosities, as can be observed in the figure. This fact supports the argument that the deviations arise from imperfections caused during the manufacturing process, where the removal of the ceramic slurry is a critical issue, particularly at lower porosities. The viscous and inertial constants f1 and f2 obtained from analyzed foams and their confidence limits are listed in Table 7. Since these constants are dependent on the fluid properties, the representative constants for a foam geometry, independently of and , are given by the viscous and inertial permeability parameters k1 and k2 , also listed in the table. Most of the authors in the literature have attempted to describe the pressure drop of open-cell foams with the Ergun equation (Richardson et al., 2000; Moreira et al., 2004; Lacroix et al., 2007), describing the pressure drop of granular beds:

p

(1 − )2 = 150 · · · U0 3 2 L  Dparticle + 1.75 ·

(1 − )

3

·

 Dparticle

· U02 .

(23)

5214

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217 Table 8 Features of the foams analyzed in the literature (pressure drop)

Table 7 Viscous and inertial constants and permeabilities for the analyzed foams 9

3

t (–)

Sgeo (m−1 )

Richardson et al. (2000) 10 1.68 30 0.826 45 0.619

0.88 0.88 0.80

2064 4032 5100

Moreira et al. (2004) 8 20 45

0.94 0.88 0.76

1098 2304 5616

Foam

f1 (Pa s/m2 )

f2 (Pa s2 /m3 )

k1 × 10 (m2 )

k2 × 10 (m)

(PPI)

10PPI_0.8 20PPI_0.8 30PPI_0.8 45PPI_0.8 10PPI_0.85 20PPI_0.85 30PPI_0.85 45PPI_0.85 20PPI_0.75 M_40PPI

941 ± 66 2933 ± 313 3719 ± 204 4315 ± 211 681 ± 65 1835 ± 124 2430 ± 298 2704 ± 240 3993 ± 398 2108 ± 105

252 ± 11 474 ± 32 546 ± 21 529 ± 21 133 ± 11 398 ± 11 419 ± 31 432 ± 25 583 ± 40 215 ± 6

28.59 9.17 7.23 6.23 39.5 14.66 11.07 9.95 6.74 12.76

3.13 1.67 1.45 1.49 5.94 1.98 1.89 1.83 1.36 3.67

Dp (mm)

2.3 0.8 0.36

18 25 20PPI_0.75 empirical model 20PPI_0.75 Lacroix model 20PPI_0.75 20PPI_0.8 empirical model 20PPI_0.8 Lacroix model 20PPI_0.8

15

14 Δp/ΔL [Pa/m]*10-4

Δp/ΔL [Pa/m]*10-4

20

PPI 45_0.85 30_0.85 20_0.85 10_0.85 Rich. 45 Rich. 30 Rich. 10 Schlegel 30 Schlegel 20 Schlegel 10 Moreira 45 Moreira 20 Moreira 8

16

10

12 10 8 6 4 2

5

0 0

3

6

9

12

15

U0 [m/s]

0 0

3

6

9

12

15

U0 [m/s] Fig. 21. Comparison of Ergun-based equation by Lacroix et al. (2007) with experimental pressure drop data and empirical relation from this work.

The dependence of the pressure drop on porosity in Eq. (23) arises from the definition of the hydraulic diameter for a packed bed of spheres: Dh = (4 · )/Sgeo = (2/3) · (/(1 − )) · Dparticle . Richardson et al. (2000) use the model of parallel cylinders for calculating the geometric surface area, introducing an additional influence on porosity, but leaving the original dependence on porosity in Eq. (23). Since their data were still not well represented by the equation (because the model is not correct), the authors calculate new constants of the Ergun equation by an empirical fit, introducing yet another dependence on the porosity and the pore diameter. Moreira et al. (2004) use the general Forchheimer equation with the dependence on porosity of the Ergun equation. The dependence of pressure drop on pore diameter was empirically fitted to their data. Lacroix et al. (2007) use the unchanged Ergun equation, substituting the particle diameter by the strut diameter derived from CC model. The substitution is based on the comparison of foams and particles with the same specific surface area per unit volume and the same porosity, leading to Dparticle = 1.5 · dt . Hence, the modified Ergun equation proposed by Lacroix et al. is a combination of two different geometric models, the hydraulic diameter model, from which the dependence of the Ergun equation on the porosity and particle diameter arises, and the CC model. The authors conclude honestly in the publication: “Although no physical reason can be invoked in principle to explain the extension of the Ergun equation, the model is in good agreement with experimental data . . . ”. The correlations from the literature were evaluated with the experimental pressure drop from this work. The predictions by all literature correlations were not satisfying. It is not surprising that different authors use different approaches for describing their pressure drop values, since the published pressure drop data of foams vary considerably, presum-

Fig. 22. Comparison between pressure drop data of this work with literature data. Different shadings represent different references; circles—10 PPI, diamonds—20 PPI, triangles—30 PPI and squares—45 PPI.

ably because irregularities in the morphology differ dependent on the foam manufacturer and the raw material used. With the experimental values for permeability (k1 , k2 ) and geometric (h , Dp ) parameters of the foams, an empirical relation was found to describe the permeability constants: k1 = 1.42 × 10−4 ·  k2 = 0.89 ·



 Dp 1.18 7.00 · h , m

 Dp 0.77 4.42 · h . m

(24)

(25)

The empirically fitted permeability parameters were compared to the model of Lacroix et al., since it displayed the closest representation of the pressure drop data in this work among all literature correlations. Fig. 21 shows that the data obtained from 20 PPI foams with two different porosities, as an example, are not well represented by the modified Lacroix–Ergun equation. For substitution of the particle diameter in the Ergun equation, Eq. (11) was taken, since using the experimental strut diameters from this work resulted in a stronger overprediction of the pressure drop than one displayed in the figure. The reason for the better representation of the experimental pressure drop as well as the specific surface area data upon using Eq. (11) for calculating the strut diameter is closely related to the fact that, due to the manufacturing process, the strut diameter does not increase proportionally to the nod size as porosity decreases. The ceramic precursor deposits more at the intersections than along the struts. This fact was also observed by Lacroix et al. (2007). Hence, taking merely the experimental strut diameter does not account for the solid deposited in the intersections, predicting a far too high geometric surface and with it, a higher pressure drop of the structure. The same conclusion is obtained from the dependence of pressure

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

10

100

45PPI_0.85 30PPI_0.85 20PPI_0.85 10PPI_0.85 45PPI_0.8 30PPI_0.8 20PPI_0.8 10PPI_0.8 20PPI_0.75

1 1.0E+03

1.0E+04

Shfoams/ (Sc1/3*FL) [-]

Sh [-]

100

1.0E+05 Hg [-]

1.0E+06

5215

Modified-Lévêque Eq. CI_Upper Limit (+20 %) CI_Lower Limit (-20 %)

10

1 1.0E+03

Sh 1/3 2.34 0.48 Sc ⋅ εh ⋅ (Dp/0.001m)

= 0.62 ⋅ Hg

0.31

1.0E+04 1.0E+05 Hg [-]

1.0E+06

Fig. 23. Mass transfer (Sh) versus pressure drop (Hg) properties. Left: raw data; right: correlated data.

drop on pore diameter, which is as well strongly overrated by the Lacroix–Ergun equation when applied to our foam samples. Since none of the correlations published are able to predict pressure drop values in a general way, there is no reason to favor any of these semi-empirical Ergun based models over the Forchheimer equation for flow through porous structures. The parameters given in Eqs. (24) and (25)give a good representation of the influence of porosity and pore diameter on pressure drop. For the reasons explained at the beginning of this section the difference between the experimental permeability parameters ki,exp and the relations given above can be up to 25%, so these equations should be used only for a first estimation of the pressure drop in foams. Nevertheless, 3–5 different pieces of the same foam type were assessed, yielding quite representative average values of the pressure drop for each foam type. In contrast, existing literature correlations are based on a rather small selection of specific foam samples.

3.3.3. Comparison with literature results To compare with literature data, only data from foams with features similar to those of our samples were taken into account. Schlegel et al. (1993) measured the pressure drop of foams with 85%. Even though the sizes of their samples were significantly different, the authors observed similar pressure drop values in the case of 20 and 30 PPI foams. Moreira et al. (2004) measured the resistance to flow of three SiC–Al2 O3 foams in water and air. The features of the foams from the literature, as given by the authors, are listed in Table 8. Whereas the pressure drop of their 8 and 20 PPI foams are qualitatively in agreement with the data from this work, the pressure drop of the 45 PPI is significantly higher (Fig. 22). The reason for this is the varying porosity of the foams from Moreira et al. While the 8 PPI foam has a porosity of 94%, the void fraction of the 45 PPI is merely 76%. This decrease in porosity could be explained by the increasing difficulty for ceramic slick removal during manufacturing, leading not only to a higher density, but also to a higher amount of closed pores, strongly influencing the permeability of the foams. The pressure drop of 10, 30 and 45 PPI Al2 O3 foams measured by Richardson et al. (2000) is plotted in Fig. 22. The authors found considerably higher values for the pressure drop of the 30 and 45 PPI foams. Since the material was Al2 O3 and the dimensions were similar, the reason for the deviations is most probably a different quality of the used structures. The amount of closed pores of a given foam sample can differ significantly, explaining the deviations between pressure drop values of different authors. Another possible reason is the systematic difference in pore diameter for foams having the same pore count, which were manufactured by different producers. The pore diameters given by these last authors are smaller than the ones from this work, as can be observed upon comparison of Tables 1 and 8.

One important conclusion from the evaluation and comparison of pressure drop data of foams from different authors is that no general relation for the prediction of permeability in foams can be yet achieved. The resulting morphology and quality of foams from different manufacturers differ considerably, despite of using the same (replication) method. Manufacture process optimization is needed for the reproducible production of foams with a constant pore density having similar permeabilities. More attention should be given to the amount of closed pores, substantially affecting pressure drop. A volume imaging method like MRI could quantify closed pores in a given sample. If the real structure of this sample could be reconstructed in a CFD grid, pressure drop could be simulated and the local pressure increases leading to higher pressure drop could be recognized, correlating the pressure drop to the geometric features of the sample, including the amount of closed pores. 3.3.4. Verification of the Lévêque analogy between mass transfer and pressure drop The analogy of transport phenomena is based on the close similarity of the phenomenon of mass, heat and momentum transfer in fluids, suggested by the fact that the basic equations describing the fluxes have the same form, making it possible to apply an analysis of one of the phenomena to the other two, when geometry and boundary conditions are equivalent. It has been of great scientific interest to develop well substantiated relations for the prediction of mass transfer from the similar processes of fluid friction and heat transfer, where there is no available data. The Chilton–Colburn analogy between mass and heat transfer is based on the “modified Reynolds analogy” between heat transfer and fluid friction (Chilton and Colburn, 1934). In contrast to the postulated heat–mass transfer analogy, the Reynolds analogy holds only for turbulent flow inside tubes and parallel to plane surfaces, but does not apply to flow around tubes or tube banks, where drag becomes decisive. An analogy between pressure drop and heat (or mass) transfer in various heat exchanger types was found by Martin (2002). It is based on the generalized Lévêque equation: Nu Pr1/3

=

  D 1/3 = 0.404 · 2xf · Hg · h , L Sc1/3 Sh

(26)

where Hg is the dimensionless form of the pressure drop, defined as Hg =

 2

· Re2 =

p D3p ·  L  · 2

(27)

and does not contain the flow velocity.  is the Darcy friction factor, xf is the frictional fraction cF /cD , Dh the hydraulic diameter and L the length in direction of flow. The original equation was derived by Lévêque (1928) for a developing thermal (or diffusive) boundary layer in a fully developed laminar flow in the entrance region of a tube. The equation can also be applied to turbulent flow, as long

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as the thermal boundary layer remains within the viscous sublayer. It has been proven for packed beds and other structures that heat (and mass) transfer coefficients can be predicted from pressure drop data with this analogy (Martin, 2002), since it holds for structures where flow is constantly disturbed, and the (thermal, viscous and diffusive) boundary layers are constantly interrupted and partially rebuilt along the direction of flow. One of the main goals of this contribution was to verify if this analogy, valid for a wide variety of spacewise geometric structures, is applicable to irregular cellular structures like foams, where the flow around the struts and vortices is repeatedly disturbed as well. The experimental Sh and Hg values of the foams reported in this work are depicted in the left plot in Fig. 23. The mass transfer coefficients of foams are proportional to the cubic root of the pressure gradient. It can be therefore concluded that the generalized Lévêque equation represents qualitatively the hydrodynamic mass transfer behavior of foams. However, the mass transfer coefficients of foams with different pore counts cannot be correlated with the pressure drop by using Hg and Sc as only dimensionless arguments. This could be expected, since the additional geometric influence of the structure was needed in order to correlate Sh with Re as well. Following the same procedure as in the parameter fitting of the Sh–Re correlation (Eq. (21)), the influence of geometry on the Lévêque analogy was considered. The right diagram in Fig. 23 shows the Sh–Hg correlation with the geometric parameter FL . It can be concluded that, by using the modified Lévêque equation given in Eq. (28), mass transfer coefficients of foams can be predicted from experimentally more accessible pressure drop data with an accuracy of ±20%  Shfoam = 0.62 ·

 0.48 Dp · 2.34 · Hg0.31 · Sc1/3 . h 0.001m

(28)

4. Conclusions The mass transfer and pressure drop properties of ceramic foams with pore densities between 10 and 45 PPI and porosities between 75% and 85% were evaluated. The dimensionless mass transfer coefficients of foams (Sherwood numbers) increase with decreasing pore count at constant hydrodynamic conditions (Reynolds numbers). In contrast to packed beds of particles, the mass transfer properties of foams cannot be completely described with the hydrodynamic and fluid properties of the system (Reynolds and Schmidt numbers) as the only influencing parameters, since the geometry of foams with different pore densities and porosities is not perfectly similar. By means of a geometric characterization of these reticulated structures, the additional influence of geometric parameters such as pore (and strut) diameter, geometrical surface and porosity on the mass transfer coefficients was evaluated as far as possible and taken into consideration. From this evaluation, a quantitative relation for the prediction of mass transfer coefficients of foams with different pore sizes and void fractions was established in a range of Reynolds between 7 and 1100 within ±9% accuracy. A metallic foam (40 PPI, stainless steel) with a porosity of 95% was also analyzed for comparison. The experimental Sh values are in excellent agreement with the values predicted by the correlation for this foam. It can therefore be concluded that the correlation can also be used for foams with higher porosity and different materials. An important prerequisite for the use of the correlation is the quantitative characterization of the pores and struts, based on statistical image analysis, as well as the knowledge of the hydrodynamic relevant void fraction of the structure, available from pycnometry methods with anon-wetting fluid. The pressure drop of foams increases with increasing pore count and decreasing porosity. Due to the manufacturing process of the foams, the samples exhibit closed pores to a certain, not quantifiable

extent, leading to considerable deviations in the pressure drop values of different samples from the same foam type, especially at high superficial velocities. The pressure drop versus velocity data follows, as expected, the Forchheimer equation. Empirical relationships were found for the permeability parameters, showing a strong influence of the void fraction. Based on the analogy between mass (or heat) transfer and pressure drop (Lévêque analogy), the mass transfer data (Sh values) could be correlated to the experimental pressure drop of the structures (Hagen values) within ±20%, confirming the possibility to use this analogy for a first (qualitative) prediction of mass transfer coefficients from pressure drop data. Notations A

R Sgeo U0 V V

pre-exponential parameter in dimensionless correlation, dimensionless exponent in Reynolds number (Re), dimensionless drag coefficient, dimensionless friction factor (< cD ), external flow, dimensionless heat capacity, J/(g K) relative correlation matrix, dimensionless inlet CO concentration, mol/m3 outlet CO concentration, mol/m3 relative covariance element (i, j) in the correlation matrix, dimensionless inner pore diameter, m strut diameter, m diameter, m hydraulic diameter, m pore diameter, m molecular diffusion coefficient, m2 /s viscous constant, Pa s/m2 inertial constant, Pa s2 /m3 dimensionless geometrical function, dimensionless dimensionless geometrical function in modified Lévêque equation, dimensionless heat transfer coefficient, W/(m2 K) Jacobian matrix, dimensionless mass transfer coefficient gas/solid, m/s volumetric rate coefficient based on the bulk volume, s−1 viscous permeability parameter, m2 inertial permeability parameter, m bed length, m parameter in Fg , dimensionless mass of the foam sample, kg parameter in Fg ,dimensionless number of experimental points for mathematical modelling, dimensionless pressure, Pa pressure drop, Pa recycle flow rate ratio, dimensionless geometric surface area per bed volume, m2 /m3 (bed) superficial velocity, m/s volume, m3 variance–covariance matrix, dimensionless

V0 xf XCO

inlet flow rate, m3 /s frictional fraction (cF /cD ), dimensionless CO conversion, dimensionless

B cD cF cp C CCO,in CCO,out Ci,j dP dt D, d Dh Dp D1,2 f1 f2 Fg FL h J km kV k1 k2 L m mfoam n N p

P

•

Greek letters

 h

surface tension, N/m hydrodynamic relevant porosity, dimensionless

G. Incera Garrido et al. / Chemical Engineering Science 63 (2008) 5202 -- 5217

t

   

total porosity, dimensionless dynamic viscosity, Pa s contact angle, dimensionless thermal conductivity, W/(m K) kinematic viscosity, m2 /s darcy friction factor, dimensionless density, kg/m3 root mean square deviation, m

Dimensionless groups D3 · Re2 = p · p2 Hg =  2 L · h·Dp Nu =

·cp Pr = U ·D Re = 0 p km ·Dp Sh = D 1,2

Sc = D

1,2

Hagen number Nusselt number Prandtl number Reynolds number Sherwood number Schmidt number

Abbreviations CC CSTR Dev GHSV MRI NDIR PFR PPI SFRD STP TTKD Var var

cubic cell continuous stirred tank reactor deviation gas hourly space velocity magnetic resonance imaging non-dispersive infrared plug flow reactor pores per inch supercritical fluid reactive deposition standard temperature and pressure tetrakaidecahedron experimental error variance variance of the parameter

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