The effective thermal properties of solid foam beds: Experimental and estimated temperature profiles

https://doi.org/10.1016/j.ijheatmasstransfer.2010.04.033Get rights and content

Abstract

The effective radial heat conductivity of a solid foam packing and the wall heat transfer coefficient are determined under fluid flow conditions typical of catalytic reactors. A detailed 2-D heterogeneous model is phase-averaged in order to rigorously define lumped heat transfer parameters. The resulting pseudo-homogeneous model involves two fitting parameters only and it is successfully compared with experiments. First, experiments with packed extrudates validate the approach in comparison with known results. A second experiment with solid foams (polyurethane and SiC) allows correlating the radial heat conductivity to the nature of the solid, its morphology and fluid flow characteristics. The method is inspired from the correlations for particles and seems very promising. Conversely, determining the wall heat transfer coefficient yields only an average value (110 W m−2 K−1 ± 15%) and correlation with fluid velocity is impossible in the studied range 0.018–0.32 m s−1.

Introduction

Packed beds are extensively used in the chemical and process industries as reactors, separators, dryers, filters and heat exchangers. The reactor packing aims to increase the rate of heat and mass transfer by increasing the gas–solid contact surface and by increasing the turbulence within the fluid phase. One relevant example is the packed bed of pellets, however, due to the low porosity (in the range of 0.3–0.6), these packed beds induce important pressure drops at high flow rates which is detrimental for the global process. The idea of moving from these traditional packed beds to the structured bed (e.g. monolith or wire), has become more and more popular. More recently, the solid foams have been introduced to overcome some of the above shortcomings of ‘conventional’ packings. Solid foams are porous materials with low densities and novel thermal, mechanical, electrical and acoustic properties [1]. In contrast to ‘conventional’ packings of granular material, the use of solid foams have become very attractive, since they offer to vary the geometry for the solid–fluid contact, and especially the bed voidage (0.70 < ε < 0.95). This new medium has as a highly permeable porous structure, which enables a considerable reduction of the pressure drops along the bed even with a high specific surface area. Solid foams have been used for a long time in the design of aircraft wing structures in the aerospace industry, core structure for high strength panels, and also in compact heat exchangers. More recently, they have been considered as potential candidates for catalytic support [2], [3], [4], [5], [6], [7].

In the operation of multitubular packed bed reactors for gas/solid catalytic processes which constitutes the focus of this work, it is well known that the removal of the reaction heat and the pressure drop are the most critical aspects. Knowing that the size (length and diameter) of the industrial reactors is strongly related to the marked temperature gradients, the value of effective radial thermal conductivity plays an important role [8]. A ‘high’ heat conduction in the catalytic solid matrix would afford reduced unacceptable hot spots in fixed-bed reactors, resulting in reduced risks of thermal runaway, in better thermal stability of the catalyst, and possibly in improved selectivities; eventually, new reactors could be designed for increased throughput and/or with enlarged tube diameter, with significant economic benefits. Because the geometry of the solid foam beds is significantly different of ‘conventional’ packed beds, there is a need for experimental and theoretical works concerning effective radial thermal conductivity in industrial conditions. If it appears today that the pressure drop for the solid foams has become largely documented in the recent literature [9], on the contrary, there are relatively few investigations of transport phenomena. In open literature, we can find different works which proposed empirical, analytical or numerical model depending on the porous morphology and on the conductivities of the fluid and solid phases to estimate the effective thermal conductivity of solid foams. These works have been summarized in review articles [10], [11]. However, to illustrate this, we can cite a non-exhaustive list of these main studies:

Calmidi and Mahajan [12] and Boomsma and Poulikakos [13] independently developed models utilising geometrical estimate for calculation of effective thermal conductivity specifically for metallic solid foams saturated with a fluid. For high porosity metal foams Calmidi and Mahajan [12] presented a one-dimensional heat conduction model considering the porous medium to be formed of a 2-D array of hexagonal cells. Whereas Boomsma and Poulikakos [13] proposed an analytical effective thermal conductivity model based on the tetrakaidecahedron cells with cubic nodes at the intersection of the struts. The results of these models are in good agreement with experimental measurements made on aluminum solid foams with air or water as the saturating fluid. They show that, despite the high porosity of the foam, the heat conductivity of the solid phase controls the overall effective thermal conductivity to a large extent, and that an accurate representation of the contribution of the solid is needed to model the effective conductivity. More recently, Bhattacharya et al. [14] also provided an analysis for estimating the effective thermal conductivity of high porosity metal foams. They represented also the open-cell structure by a model consisting of a 2-D array of hexagonal cells. The presence of lumps of metal at the junction of two struts is taken into account by considering square or circular blobs of metal, which results in a sixfold rotational symmetry. The analysis shows that the porosity and the ratio of the cross-sections of the struts and the intersection strongly influence the results but that there is no systematic dependence on the size of the cells. Both models involved a geometric parameter that was evaluated using the experimental data. It is interesting to note that these works are based upon the original work done by Zehner and Schlünder [15] on packing of spheres. Indeed, the authors divided the unit cell in several layers in series, each layer containing solid and fluid phases in parallel. Finally, the effective conductivity is usually splitted into a static (or stagnant) contribution and a flow contribution (or dispersion conductivity, λra,effgd), and several empirical and analytical studies have attempted to quantify dispersion in porous and fibrous media [16], [17], [18], [19]. The general conclusion is that λra,effgd is linearly proportional to the local flow velocity and becomes prominent at high Reynolds numbers, especially if the stagnant effective conductivity is small in magnitude. In case of forced convection in solid foams, a detailed study (experimental and numerical methods) has been performed by Calmidi and Mahajan [20]. The latter authors confirmed that the previous results for the ‘conventional’ packed beds remain valid for solid foams. However, because these authors used metal foams with air, the transport enhancing effect of thermal dispersion is extremely low due to the relatively high conductivity of the solid phase. The authors conclude honestly, that more studies using solid foams made of other materials is then necessary to get a better understanding of some relations.

In this context, replacement of ‘conventional’ packings with solid foam catalyst supports in multitubular gas/solid reactors can be a good potential for improvement of the heat exchange. The above consideration have prompted us to investigate the thermal characteristics of novel catalysts made of solid foams, in view of their use in externally cooled multitubular fixed-bed gas/solid reactors for strongly exothermal processes. First, the experimental method used to obtain comprehensive information of temperature field in the interior of ‘conventional’ packed beds and solid foam beds under forced convection (air flow) including both radial and axial temperature distributions is described. Next, we developed a new pseudo-homogeneous model in order to take into account the coupling between the solid and fluid phase energy equations and thus the interfacial heat transfer. The measured information and the pseudo-homogeneous model are then used to derive the effective thermal parameters. Finally for solid foam beds, simple correlations are proposed for the effective thermal dispersion.

Section snippets

Characteristics and morphological parameters of catalytic supports

Today, catalytic solid foams may be produced in a large variety of materials (Al2O3, cordierite, aluminum, copper etc.). However, from a catalytic point of view, these foam structures have a relatively low specific surface area (m2/g) due to the high temperature methods of preparation [21], [22] for performing good anchorage and dispersion of the active phase. Solid β-SiC foam (studied in this work) with a medium specific surface area and a natural wash-coat layer, i.e. SiO2 and SiOxCy topmost

Development of 2-D pseudo-homogeneous model

The aim of this work is to estimate the effective thermal properties of solid foam beds in order to correlate the radial heat conductivity to the nature of the solid, its morphology and fluid flow characteristics. In this context, a detailed heterogeneous (HT) two-temperature model is attractive since it seems ‘more realistic’ than a pseudo-homogeneous (PS) one-temperature model. According to Calmidi and Mahajan [20] the HT model is unavoidable in foam structures. Moreover, when a strongly

Method for solution

Let us recall that the main goal is to obtain the radial heat transfer coefficients from the experimental temperature curves using the pseudo-homogeneous model. This model involves three parameters, namely hintsg,λax,effsg,λra,effsg, of which the second lumps together true axial heat conduction and solid–fluid heat transfer. According to Giani et al. [5], Hsu and Cheng [39] and Sozen and Vafai [40], we consider that the axial thermal conductivity (λax,effs) can be neglected, and thus the Eq.

Conclusion

The pseudo-homogeneous 2-D model is able to describe the temperature profiles in porous media made of either packed particles or foams. It is the necessary compromise between detailed and phase-averaged descriptions that allows reliable determination of a restricted set of lumped heat transfer parameters and avoids cross-correlations among too many sub-parameters. The lumped thermal parameters are obtained from a more detailed heterogeneous 2-D model which is more physically based. This method

Acknowledgements

The authors would like to thank Total S.A. and SiCat Companies for financial and technical support.

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