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Ind. Eng. Chem. Res. 2005, 44, 9078-9085
MATERIALS AND INTERFACES Heat Transfer Characterization of Metallic Foams Leonardo Giani, Gianpiero Groppi, and Enrico Tronconi* NEMAS, Centro di Eccellenza per l’Ingegneria dei Materiali e delle Superfici Nanostrutturate, Dipartimento di Chimica, Materiali ed Ingegneria Chimica “G. Natta”, Politecnico di Milano, Piazza Leonardo da Vinci 32-20133 Milano, Italy
Gas-solid heat transfer coefficients were determined in open-celled metal foams as part of a study aimed at evaluating the application of metal foams as catalyst supports in gas-solid catalytic processes with short contact times and high reaction rates, typically controlled by diffusional mass transport. Examples of such processes are found in the field of environmental catalysis, including, for example, catalytic combustion, selective catalytic reduction of NOx by NH3 (SCR-DeNOx), automotive exhaust gas after treatment, and also in the catalytic partial oxidation of hydrocarbons for syngas or H2 generation processes. In this work, foam samples made of FeCrAlloy and Cu with nominal porosities of 10 and 20 pores per inch (ppi) were characterized by performing non-steady-state cooling measurements. Convective gas-solid heat transfer coefficients were determined by applying a one-dimensional, heterogeneous model of the cooling structure. The correlation Nu ) 1.2Re0.43Pr1/3 well described the dependence of the dimensionless heat transfer coefficients on Re and Pr numbers for all the tests made in a range of flow superficial velocities from 1.2 to 5.7 m/s, independently from the foam cell size (20 < Re < 240). Such a correlation was derived assuming a prismatic idealization of the unit cell and selecting the equivalent strut diameter as the characteristic size of the foams. This expression satisfies the Colburn analogy with the correlation for mass transfer coefficients derived in a previous investigation and resembles semitheoretical literature correlations for heat transfer in flow across banks of tubes at low Reynolds numbers. 1. Introduction work1
we estimated gas-solid mass In a previous transfer coefficients in open-celled metal foams2,3 by performing the catalytic oxidation of CO under diffusional control in a microreactor and by measuring the apparent rate constant. Foams of different pore densities were activated by deposition of a thin layer of palladium on alumina.4 On the basis of a simple geometrical description of the metallic foam as a cubic framework of connected struts, all mass transfer data were successfully correlated by the following expression:
Sh ) 1.1Re0.43Sc1/3
(1)
using the diameter of the struts as the characteristic length in dimensionless numbers. Equation 1 is useful in view of the application of metal foams as catalyst supports for fast, diffusion-controlled reactions. In many practical catalytic applications dealing, for example, with highly exothermic reactions, however, engineering calculations would call for the evaluation of interphase heat transfer, too. Besides, open-celled metal foams can be used also as highly efficient heat exchangers, taking advantage of the large exchange surface area per unit volume and of the * To whom correspondence should be addressed. Tel.: +39-02-2399 3264. Fax: +39-02-7063 8173. E-mail:
[email protected].
tortuous flow paths through the porous matrix that promotes turbulence and enhances interphase heat transfer rates. In this way, heat can be removed from or added to gases or liquids by letting them flow through the foam and cooling or heating at the same time. An example of such applications is the use of metal foams as compact heat sinks for cooling of microelectronic devices such as computer chips or power electronics.5 Lu et al.6 developed a theoretical model to evaluate the overall heat transfer coefficient, considering both conduction through the solid and heat exchange from the solid surface to the gas. Foams were modeled as simple cubic unit cells consisting of heated slender cylinders, making use of the analogy between flow through the foam and flow across a bank of cylinders. Heat transfer coefficients were determined experimentally by Younis and Viskanta7 for ceramic foams (alumina and cordierite) with pore diameters in the range of 0.29-1.52 mm. In their work, the authors estimated a volumetric heat transfer coefficient because the surface area per unit volume of the foams was unknown. A different dimensionless correlation with expression Nu ) CRem was obtained for each sample analyzed, with the pore diameter chosen as the characteristic length to calculate the Nusselt (Nu) and Reynolds (Re) numbers. The Reynolds exponent m was reported to vary in the range 0.42-0.96, while the Reynolds coefficient C took values in the range 0.139-0.638. The importance of an accurate determination of the geometrical param-
10.1021/ie050598p CCC: $30.25 © 2005 American Chemical Society Published on Web 11/01/2005
Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 9079 Table 1. Geometric Properties of the Metal Foam Samples Investigated in This Work
samplea A B C
material
cell density (%) (ppi)
pore size a (m)
Fe-Cr alloy 4.7 × 10-3 92.7 Fe-Cr alloy 2.0 × 10-3 93.7 copper 4.6 × 10-3 91.1
5.4 12.8 5.6
ds (m)
Sv (m2/m3)
0.82 × 10-3 0.33 × 10-3 0.80 × 10-3
352 767 449
a Foams A, B, and C correspond respectively to foams B, D, and F in ref 1.
Considering this model, the cell volume Vo can be calculated as
Vo ) a2b
Figure 1. Prismatic cell model for anisotropic foams.
eters of the foam was emphasized by showing that the difference in the pore size declared by the manufacturer and the measured value resulted in two different correlations for the same foam. Schlegel et al.8,9 have also determined heat transfer coefficients experimentally in ceramic foams with pore diameters from 1.02 to 5.20 mm (nominal porosities of 10, 20, 30, and 50 ppi). The identified dimensionless correlations, which took into account the pore diameter as the characteristic length, differed depending on the sample as in the work of Younis and Viskanta.7 Reported Reynolds exponents were in the range of 0.420.47 for 10, 20, and 30 ppi samples and 0.27 for a 50 ppi sample, while C coefficients varied from 0.801 to 1.6. Geometrical data were provided by the producer or measured by the authors. In this work, heat transfer coefficients have been investigated by performing non-steady-state cooling temperature measurements of hot structures in an air stream. The experimental results have been analyzed by means of a transient mathematical model of the gassolid heat transfer process in order to derive a general correlation for gas-solid heat transfer coefficients. Finally, results of both the heat and the mass transfer investigations are herein discussed comparatively in order to check their consistency with the Colburn analogy in reticulated foam structures. 2. Experimental Section 2.1. Metal Foams. Samples used in this research comprised metal foams of two different nominal cell sizes (10 and 20 ppi) and were provided by Porvair as panels of 5% nominal relative density. The foams were characterized by measurements of the pore size, by image analysis, and by estimation of the open volume fraction , calculated from the apparent density of the foam (FFOAM) and the density of the foam struts (FS) as described in more detail elsewhere.1 Adopting the same approach developed in the previous mass transfer investigation,1 isotropic foams were described according to a cubic cell model of the structure6 from which it is possible to estimate the average struts diameter ds and the foam specific geometric area Sv as functions of the pore size a and of the porosity . However, also anisotropic foams have been herein investigated. In the case of anisotropic foams, pore sizes differ when either a transversal section or a longitudinal section is considered. Accordingly, in this work the foam characteristic dimensions were derived by modifying the cubic model into a prismatic one as depicted in Figure 1.
(2)
The overall volume of the struts Vs can be expressed as a function of the foam void fraction :
Vs ) (1 - )Vo ) (1 - )a2b
(3)
Alternatively, Vs can also be calculated as the overall volume of the cylinders included in the unit cell, considering that each strut is shared among four cells:
dS2 (2a + b) 4
Vs ) π
(4)
By combining eqs 3 and 4, one obtains an expression that relates the strut diameter ds to a and :
[
dS ) 2a
b (1 - ) π (2a + b)
]
1/2
(5)
The specific area per unit cell volume is then computed according to eq 1. The specific area per unit cell volume Sv can be eventually expressed by consideration of eq 5 as a function of the foam porosity and of the pore diameter a:
Sv )
[
]
4(1 - ) 2 (2a + b)(1 - ) ) π dS a b
1/2
(6)
For isotropic foams, b equals a and the above equations reduce to those presented in ref 1. Estimates of the foam samples geometrical parameters can be found in Table 1. As reported, samples A and B were made of FeCr alloy while sample C was made of copper. The unit cell for sample C is anisotropic: its pore size is a ) 4.6 × 10-3 m in the transverse direction (defined with respect to flow) and b ) 3.3 × 10-3 m in the longitudinal direction. 2.2. Heat Transfer Runs. The present study was carried out in the same test rig used by Brautsch et al.10 to determine heat transfer coefficients in metal honeycomb-type structures at high flow rates. Cylinders of 75.0 mm diameter were cut from the metal foam panels by electroerosion, producing a precise cut with no damage for the structure. The depth of the cylinders was 25.3 mm for the Fe-Cr alloy samples A and B and 50.0 mm for the Cu sample. Tests were made over each single structure and also placing more structures in series inside the test rig, schematically depicted in Figure 2. Air was fed by a blower at a constant flow rate in the range of 5.5-25.2 L/s (STP). Hydrogen was fed from an independent line with two valves, one on/off suitable to
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Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 Table 2. Experimental Cases Studied sample A A B C
Figure 2. Schematic diagram of the test rig for heat transfer runs.
turn off fast the hydrogen feed and the other one suitable to regulate the flow rate. A mass flow meter was placed in the hydrogen line to control the flow and therefore, the composition of the air-hydrogen mixture in order to work below the lean explosion limit. Both gases were mixed in a static mixer (model SMV provided by Sulzer) before entry into the test module. In this section the mixture was ignited by a palladium catalyst supported on a cylindrical metal foam 5 mm in depth, which acted as a preheater with a fast response to changes in the hydrogen supply. The heated air flow passed through a wire screen consisting of five layers of stainless steel (mesh 100), which ensured plug flow conditions and minimized radiation effects that could affect the temperature reading of the thermocouples placed further downstream. The temperature of the air flow in the inlet section (Tin) was measured immediately prior to the foam by a bundle of five Cr-Ni alloy type K thermocouples with 0.5 mm sheath diameter and exposed tips (tip diameter ) 0.1 mm) to ensure quick response time. They were located along one diameter of the tube section in a central region of 30 mm where the influence of the tube wall was negligible, as shown schematically in Figure 2. The temperature of the gas flow at the outlet section (Tout) was measured by an identical bundle of five thermocouples. Neither the inlet bundle of thermocouples nor the outlet one were in contact with the structure and measured the temperature of the gas stream. Another wire screen was placed further downstream. The foam structure was wrapped with a thin fiber mat to prevent flow bypass. Such a fiber mat was assumed to have negligible influence on the heat transfer measurements in the structure due to its low thermal capacity. For a given air flow, the hydrogen feed was increased until the inlet temperature Tin reached about 300 °C. This temperature ensured limited heat losses to the surroundings through convection and radiation. After steady-state conditions were reached, corresponding to equal inlet and outlet gas stream temperatures, the hydrogen supply was suddenly turned off, extinguishing
length (m) 10-2
2.53 × 2 × (2.53 × 10-2) 2.44 × 10-2 5.00 × 10-2
Re(ds) 48-239 48-239 19-95 48-237
the combustion over the palladium catalyst and thus letting cold air flow through the structure. The temperature readings from the two bundles of thermocouples were then recorded until the system reached laboratory temperature. Brautsch et al.10 reported that gas-solid heat transfer coefficients in ceramic honeycomb monoliths measured in the same apparatus during calibration runs deviated less than 3% from well-established literature correlations. In this work heat transfer experiments were run on each foam sample for eight different flow superficial velocities: 1.2, 1.9, 2.5, 3.2, 3.8, 4.4, 5.0, and 5.7 m/s. The cases analyzed are reported in Table 2 where the Reynolds number is referred to the struts diameter ds as defined in the Nomenclature section and on the superficial empty tube velocity u. 2.3. Mathematical Model. Heat transfer between the foam structures and the fluid was described by a dynamic, heterogeneous two-phase one-dimensional (1D) model used by Schlegel et al.8,9 and Brautsch et al.10 On the basis of the experimental setup shown in Figure 2, the model assumed plug flow conditions, with the fluid temperature (Tf) and the solid temperature (Ts) varying only in the flow direction (1D model). In view of the good insulation of the experimental test rig, which allowed us to closely approach adiabatic conditions, a constant radial temperature distribution (Ts) in the metal foam was assumed. Adiabatic conditions were experimentally checked at steady state: indeed, temperatures measured along the radial direction and between the average foam inlet and outlet temperature of the gas differed by less than 3 °C. The following energy balances for the fluid (eq 9) and for the solid (eq 10) described the heat exchange between solid and fluid phases:
∂Tf Sv ∂Tf ) -v +h (T - Tf) ∂t ∂x Ffcf s
(7)
Sv λ s f ∂2 Ts ∂Ts ) -h (Ts - Tf) + ∂t Fscs ∂x2 (1 - )Fscs
(8)
The balance equations account for (a) the thermal mass of both the gas and the solid phase; (b) convective transport in the gas phase; (c) gas-solid heat transfer, with all the resistance assumed to be confined in the fluid film near the struts surface; (d) axial heat conduction in the solid phase. This latter term includes a factor f ) 0.4 in order to take into account the tortuous path of the conductive heat flux through the reticulated structure. The estimate of f has been computed according to a correlation by Fourie and DuPlessis11 for the effective thermal conductivity in isotropic metal foams, omitting the negligible contribution of the coupled fluid conductivity. Thermal diffusion in the axial direction in the gas phase was neglected (Pe . 1) and radiation effects were not considered since the temperature was kept below 300 °C in all tests.
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The following initial and boundary conditions were considered: At t ) 0, solid and fluid were assumed to be at the same constant temperature To corresponding to the inlet and outlet temperature measured before switching off the H2 feed valve.
Tf ) TS ) T0
[t ) 0, 0 < x e L]
(9)
At the inlet cross section of the foam structure, the fluid temperature corresponded to the experimental inlet temperature profile f(t), described by cubic spline interpolation:
Tf ) f(t)
[x ) 0, t > 0]
(10)
Figure 3. Typical experimental temperature evolution of the gas phase at the inlet and the outlet of the foam.
No flux boundary conditions for the solid were assumed:
∂TS )0 ∂x
[x ) 0]
(11)
∂TS )0 ∂x
[x ) L]
(12)
The heat transfer coefficient h in eqs 7 and 8 was obtained from the Nusselt number defined in eq 13 with the strut diameter as the characteristic length:
Nu )
hds λf
(13)
Based on the analogy with heat transfer in bundles of tubes, the following correlation was tried for Nusselt number calculation: m
Nu ) CRe Pr
1/3
(14)
where the coefficient C and the exponent m are adaptive parameters depending on the flow type and on the geometrical characteristics of the tested foams. The estimates of these two parameters were obtained by global nonlinear regression on the experimental temperature profiles of the 32 runs covering all the structures and flow rates studied. For this purpose we used a robust multimethod nonlinear regression routine developed by Buzzi-Ferraris.12 At each iteration of the regression routine, the model PDES, eqs 7-14, were solved numerically by the method of lines, involving axial discretization by backward finite differences and integration in time of the resulting ODE system by Gear’s method.13 Since the temperature varied almost 300 °C during each experiment, the physical properties of the fluid phase (air density, heat capacity, viscosity and thermal conductivity) were regarded as temperature-dependent, whereas the properties of the solid phase (either Fe-Cr alloy or copper) were averaged between room temperature and 300 °C. Results Foam samples A, B, and C were tested in the heat transfer test rig at different flow rates. As an example, Figure 3 shows typical cooling curves obtained for type A foam with a flow superficial velocity of 4.4 m/s. At the beginning, the same inlet and outlet temperature of around 300 °C was measured by both sets of thermocouples, confirming that practically adiabatic
Figure 4. Inlet and outlet temperature profiles obtained with one (b) and two (9) segments of foam type A.
steady-state conditions were reached. Once the hydrogen feed was shut down, cold air entered the structure, which resulted in a sudden drop of the air inlet temperature, as shown in Figure 3. Such a steplike inlet temperature profile could be obtained because of the low thermal mass of the catalytic foam and of the wire screen placed upstream from the first bundle of thermocouples: accordingly, the response time of the rig was negligible as compared to the dynamics of the heat transfer in foams. The structure exchanged heat with the air stream and higher gas temperatures were obtained at the foam outlet until gas-solid thermal equilibrium conditions were achieved and the inlet and outlet temperatures of the gas stream were the same within experimental error. The relatively high thermal mass of the investigated foams made the cooling transient long enough to obtain a well-resolved outlet temperature profile, quite distinct from the inlet profile. Accordingly, gas-solid heat transfer coefficients, which play a key role in the cooling transient, could be accurately estimated from the experimental data. Figure 4 compares the temperature profiles obtained at a given flow rate testing one and two in series segments of foam type A. While the evolution of the inlet temperature does not change, as the heat capacity of the upstream section of the rig is always the same, the cooling rate markedly decreases upon increasing the number of foams as an effect of the doubled thermal mass. Figure 5 illustrates the effect of the air flow rate on the temperature evolution for the two different foam samples A and B. At low flow rate (Figure 5a,d) the foam cooling transient is quite slow: as a result, the outlet temperature takes from 15 to 25 s to reach the final
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Figure 5. Inlet and outlet temperature profiles obtained for type A foams (panels a-c) and type B foams (panels d-f) at different inlet flow velocities and fitting curves.
steady-state conditions at room temperature depending on the foam geometry. Shorter cooling time intervals are required upon increasing the flow rate (Figure 5c,f). In particular, when the flow rate is increased from 1.3 to 5.0 m/s, the cooling time interval decreased by a factor of about 3. Such an effect is consistent with the larger heat capacity of the cooling air associated with the higher flow rate. Besides, a higher gas velocity results in higher heat transfer coefficients, which further increase the cooling rate. Figure 5 also illustrates the effect of the foam geometry. Foams A and B have similar thermal mass, as the heat capacity of foam A is just 15% higher than that of foam B; nevertheless, when the cooling curves recorded for both foams at the same flow rates are compared, the cooling rate is clearly faster in the case of foam B with higher cell density. This is mainly due to the higher specific surface of foam B (Table 1), which enhances the gas-solid heat transfer efficiency. Also in
this case, the increment of the heat transfer coefficient on decreasing the foam strut diameter further enhances the cooling rate of foam B. Figures 4 and 5 also show fitting curves obtained by running a global nonlinear regression on all the 32 runs herein performed. The following correlation has been obtained for Nu calculation:
Nu ) 1.2Re0.43Pr1/3
[20 < Re < 240]
(15)
The percent mean error of the regression was 6.19% and the correlation index was 0.9989. The experimental results were described accurately as apparent from the good fit obtained independently from foam type and air velocity. A slightly delayed cooling is predicted for the two foams in series (Figure 4), possibly due to an enhanced role of heat dissipation caused by the longer time interval and by the higher external surface. Indeed, even stronger deviations were noted in other experi-
Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 9083
comparing simulations generated either maximizing (f ) 1) or minimizing (f ) 0) the second term on the right-hand side of eq 8. Differences with the outlet T-profiles calculated assuming f ) 0.4 were limited to about 3 °C at the worst, and generally much less, even for the run over the copper foam C at the lowest flow rate, where the role of heat conduction would be expected to be greatest. Accordingly, we conclude that axial heat conduction played a negligible role in our experiments, and the foam cooling transients were governed essentially by interphase heat transfer. Discussion Mathematical model analysis of transient cooling experiments allowed us to obtain a correlation for the calculation of heat transfer coefficients that covers a gas velocity range from 1.2 to 5.7 m/s in metal foams with measured pore sizes from 2.0 to 4.7 mm and void fractions from 91.1 to 93.7, corresponding to a 20-240 range of Reynolds numbers defined with the diameter of the struts ds as the characteristic length and the superficial velocity at empty tube. Notably the Nu correlation (eq 15) is quite similar to the one found for mass transfer in metal foams in a previous investigation based on CO combustion tests over Pd/Al2O3 washcoated metallic foams1:
Sh ) 1.1Re0.43Sc1/3
Figure 6. Inlet and outlet temperature profiles obtained for type C, anisotropic copper foam and fitting curve for different flow velocities: (a) 2.5 m/s; (b) 4.4 m/s; (c) 5.7 m/s.
(1)
The consistency of eqs 15 and 1 with the Colburn analogy provides further confidence in the results. Notice that the correlations were obtained independently and with totally different methods: mass transfer coefficients were determined in tests where the catalytic oxidation of CO was performed in a diffusioncontrolled regime on coated foams, while heat transfer coefficients were determined by performing non-steadystate cooling measurements over uncoated samples. The agreement between the results of heat and mass transfer studies also confirms the good quality of the washcoating technique developed to produce the catalytically active foams4 used in CO combustion experiments. Moreover, the higher flow rates achieved in the heat transfer tests allow to extend the validity of the mass transfer correlation (eq 1) to higher Reynolds numbers. On a more general basis, it is worth emphasizing that eq 15 closely resembles semitheoretical literature correlations for heat transfer in flow across banks of tubes at low Reynolds numbers:14
Nu ) 0.9Re0.4Pr1/3 ments with three foams in series, which were therefore not considered in the final analysis and are not reported herein. The same explanation may apply also to the overestimation of the outlet T-profile apparent in Figure 5 for case d only, which corresponds to the experiment at the lowest flow velocity over the foam sample B associated with the highest specific surface. As shown in Figure 6, where experimental and calculated curves obtained for foam C are reported, the model herein developed accurately describes also the behavior of the anisotropic copper foam, proving that a simple modification of the cubic cell model is able to catch the geometric characteristics of such anisotropic foams. A simple sensitivity analysis concerning the influence of axial heat conduction in the foams was conducted by
[10 < Re < 100]
[10 < Re < 100]
(16)
Note that the literature correlation (eq 16) was derived on the basis of the interstitial velocity at the restricted section of the bundles of tubes for Re calculation, while in this paper, the superficial gas velocity at empty tube was used. Considering the geometry of the investigated foams, the maximum interstitial gas velocity through the reticulated structure is 1.4 times higher than the empty tube velocity. Accordingly, for fair comparison with the literature correlations, coefficients in eqs 15 and 1 should be corrected to 1.0 and 0.9, respectively, which closely resemble the coefficient in eq 16. However, since the void fractions of all the foams samples investigated herein and in our previous mass transfer work1 are very close, our data do not allow us to identify
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correlation to higher Reynolds numbers (10 < Re < 240). The derived correlation also shows a strong similarity with literature expressions for heat transfer in tube bundles, which validates the concept of modeling the foam as a array of cylinders. The experiments performed in this work do not allow us to discriminate between the use of either the superficial velocity (u) or the maximum velocity (umax) to correlate heat transfer data, as all the investigated foam samples had approximately the same void fraction. To clarify this point, experiments with foams with significantly lower void fractions ( ) 0.80) will be performed in future investigations. Figure 7. View of the foam structure.
Acknowledgment the best criteria for the correlation of the experimental results with respect to flow velocity. The results herein reported also closely resemble those obtained by Satterfield and Cortez15 for mass (and heat) transfer in woven-wire screen catalysts (gauzes). On the basis of mass transfer experiments results and on critical reevaluation of previous literature data, these authors clearly point out that for high-porosity screens, a model of parallel wires normal to the direction of flow would seem more physically realistic than a porous solid or packed-bed model. Similarly, our results confirm that the description of foam structures consisting of a connected network of struts (Figure 7) based on the analogy with bundles of tubes is more physically sound than the one based on the analogy with packed beds of particles previously proposed in the literature.16 More complex and, possibly, more accurate geometric models than the simple cubic cell can be adopted, namely, for example, the tetrakaidecaedron cell.17-20 However, the increasing complexity of the model seems contradictory with the relatively poor accuracy in determining key geometric parameters such as the diameter of the pores and of the struts in the foam structure. Besides, it has been proven in this paper that a simple geometric model allows us to handle the foam anisotropy simply by switching from a cubic to a prismatic description of the unit cell. Conclusions Gas-solid heat transfer coefficients were measured in FeCrAlloy and Cu foams of different pore sizes by performing transient cooling experiments. The cubic cell model previously used to describe the foam network of connected struts and to estimate the strut equivalent diameter in isotropic samples was extended to characterize anisotropic samples. In this case the cell is represented as a prism, and expressions to estimate the surface area and the cylinder diameter were derived. By use of the equivalent strut diameter as the characteristic length, a single correlation was obtained, unifying all the tests for isotropic and anisotropic foams and thus confirming the accuracy of the modified cell model in the estimation of the foam geometrical properties. Furthermore, the correlation satisfies the Colburn analogy with the one for interphase mass transfer derived in our previous work,1 confirming the good quality of the coating used in the mass transfer tests to activate the foam surface. Moreover, the correlation herein derived extends the validity range of the previous
We gratefully acknowledge the help provided by Timothy Griffin, Dieter Winkler, and Regina Granacher from Alstom Power Technology Center, Baden-Da¨ttwill, Switzerland, where the heat transfer measurements herein reported were performed. Nomenclature a ) cubic cell dimension equal to the pore diameter (m) b ) cubic cell depth for anisotropic foams (m) cf ) specific heat of gas phase (J‚kg-1‚K-1) cs ) specific heat of solid phase (J‚kg-1‚K-1) C ) Reynolds coefficient in heat transfer correlation D ) diffusivity (m2 s-1) dS ) equivalent diameter of foam struts in the cubic unit cell (m) f ) effective heat transfer conductivity factor in foams h ) heat transfer coefficient (W‚m-2‚K-1) kmt ) mass transfer coefficient (m‚s-1) L ) depth of foam sample (m) m ) Reynolds exponent in the heat transfer correlation Nu ) Nusselt number based on struts equivalent diameter [Nu ) hdS λ-1] Pe ) Peclet number [Pe ) RePr] ppi ) foam pore density (pores per inch) Pr ) Prandtl number [Pr ) cf µ λ-1] q ) heat flux (W‚m-2) Re ) Reynolds number based on strut equivalent diameter [Re ) FdSu µ-1] STP ) standard temperature (273 K) and pressure (1atm) Sc ) Schmidt number [Sc ) µ F-1 D-1] Sh ) Sherwood number [Sh ) kmt ds D-1] Sv ) external surface area per unit volume of bed (m2 ‚m(bed)-3) t ) time (s) Tin ) inlet temperature (°C, K) Tout ) outlet temperature (°C, K) Ts ) solid temperature (°C, K) Tf ) fluid temperature (°C, K) T0 ) initial foam temperature (°C, K) u ) superficial flow velocity (m‚s-1) v ) interstitial flow velocity (m‚s-1) Vo ) foam volume (m3) Vs ) solid volume (m3) W ) foam mass (kg) w/w ) weight percentage x ) axial coordinate (m) Greek Symbols ) foam porosity λ ) thermal conductivity (W‚m-1‚K-1) µ ) gas viscosity (kg‚m-1‚s-1) Ff ) gas density (kg‚m-3)
Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 9085 FS ) apparent density of the hollow struts [FS ) W/Vs] (g‚m-3)
Literature Cited (1) Giani, L.; Groppi, G.; Tronconi, E. Mass transfer characterization of metal foams. Ind. Eng. Chem. Res 2005, 44, 49935002. (2) Gibson, L. J.; Ashby, M. F. Cellular Solids; Pergamon Press: Oxford, U.K., 1988. (3) Ullmann’s Encyclopedia of Industrial Chemistry, 6th ed.; Wiley: Weinheim, Germany, 2003; p 21. (4) Giani, L.; Cristiani, C.; Groppi, G.; Tronconi, E. Washcoating method for Pd/γ alumina deposition on metallic foams. Appl. Catal. B: Environ., in press; published online August 26, 2005, 10.1016/ j.apcatb.2005.07.003. (5) Banhart J. Manufacture, characterisation and application of cellular metals and metal foams. Prog. Mater. Sci. 2001, 46, 559-632. (6) Lu, T. J.; Stone, H. A.; Ashby, M. F. Heat transfer in opencell metal foams. Acta Metall. 1998, 46, 3619-3635. (7) Younis, L. B.; Viskanta, R. Experimental determination of the volumetric heat transfer coefficient between stream of air and ceramic foam. Int. J. Heat Mass Transfer 1993, 36, 1425-1434. (8) Schlegel, A.; Benz, P.; Buser, S. Wa¨rmeu¨bertragung und Druckabfall in keramischen Schaumstrukturen bei erzwungener Stro¨mung, Wa¨rme- Stoffu¨bertrag. 1993, 28, 259-266. (9) Schlegel, A.; Benz, P.; Buser, S.; Bestimmung der Wa¨rmeu¨bergangskoeffizienten in keramischen Schaumstrukturen, Technische Mitteilung (TM-51-91-05), Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland, 1991. (10) Brautsch, A.; Griffin, T.; Schlegel, A. Heat Transfer characterization of support structures for catalytic combustion. Int. J. Heat Mass Transfer 2002, 45, 3226-3231.
(11) Fourie, J. G.; Du Plessis, J. P. Effective and Coupled Thermal Conductivities of Isotropic Open-Cellular Foams. AIChE J. 2004, 50, 547-556. (12) Donati G.; Buzzi-Ferraris G. Powerful method for HougenWatson Model Parameter Estimation with integral Conversion Data. Chem. Eng. Sci. 1974, 29, 1504-1509. (13) Hindmarsh, A. C. Odepack, a systematized collection of ODE solvers in scientific computing, Stepleman, R. S., et al., Eds.; North-Holland: Amsterdam, 1983; pp 55-64. (14) Incropera, F. P.; DeWitt, D. P. Introduction to Heat Transfer; J. Wiley: New York, 1996. (15) Satterfield, C. N.; Cortez, D. H. Mass Transfer Characterization of Woven-Wire Screen Catalysts. Ind. Eng. Chem. Fundam. 1970, 9, 613-620. (16) Richardson, J. T.; Peng, Y.; Remue, D.; Properties of ceramic foam catalyst supports: pressure drop. Appl. Catal. A: Gen. 2000, 204, 19-32. (17) Gibson, L. J.; Ashby, M. F. Cellular Solids; Pergamon Press: Oxford, U.K., 1988. (18) Boomsma, K.; Poulikakos, D. On the effective thermal conductivity of a three-dimensionally structured fluid-saturated metal foam. Int. J. Heat Mass Transfer 2001, 44, 827-836. (19) Fourie, J. G.; DuPlessis, J. P. Pressure drop modelling in cellular metallic foams. Chem. Eng. Sci. 2002, 57, 2781-2789. (20) Kwon, Y. W.; Cooke, R. E.; Park, C. Representative unitcell models for open cell metal foams with or without elastic filler. Mater. Sci. Eng. A 2003, 343, 63-70.
Received for review May 20, 2005 Revised manuscript received September 5, 2005 Accepted September 28, 2005 IE050598P
