Permeability Of Foams Research Articles

Exploring the design space of the effective thermal conductivity, permeability, and stiffness of high-porosity foams

With knowledge of only a few effective properties of a porous structure, the applicability of the structure for a given system can quickly be determined. This study numerically simulates the effective thermal conductivity, permeability, and stiffness of high porosity structures. Commonly used isotropic architected porous structures are compared with commercially available stochastic metal foams. The architected structures include body-centered cubic (BCC) lattice, shell-based triply periodic minimal surface (TPMS), and hybrid foam (HF) composed of beam and shell with multiple adjustable parameters. The simulated effective properties touch on the applicability in heat transfer, fluid flow, and mechanically stressful situations. The dimensionless effective properties of the structures are presented in graphical form to clearly illustrate structurally dependent properties. Compared to the stochastic metal foams, the architected structures (BCC, TPMS, and most HF) showed higher effective thermal conductivities and permeabilities. This indicates a potential to improve the efficiency of a thermal or fluid flow system by replacing the stochastic foam with architected foam. Additionally, the HF structure shows broad tunability of specific properties. All effective properties simulated were rendered dimensionless to only reflect the impact of topology, and plotted in charts to show trends. These charts can aid in the selection of porous structures in diverse applications.

Conductivity and permeability of graphite foams: Analytical modelling and pore-scale simulation

Graphite foams with excellent effective thermal conductivity and large surface area have primarily constituted a new area for the emerging fields of energy conversion, conservation, and management, potentially crucial for reaching the goals of carbon neutrality and emission peak. For such energy applications, graphite foam's thermal conductivity and permeability pave the physical foundation for understanding, designing, and operating thermofluidic flows inside the porous medium. However, previous prediction models of conductivity and permeability seldomly considered the effects of random distributions of pore shape and size intrinsically induced during processing of the graphite foam. To rectify this problem, analytical models of permeability and effective thermal conductivity for graphite foam are derived based on fractal theory, being duly accounted for random distributions of pore shape and size. In parallel, pore-scale numerical simulations are carried out, providing cross-validation and shedding light on transport mechanisms at pore level. Analytical model predictions and numerical simulation results are compared with existing experimental data. Results revealed that fractal analytical models accurately predicted the permeability and conductivity of graphite foams in a porosity range from 0.686 to 0.918, with different parent ligament materials and filling fluids (e.g., air and paraffin wax).

Analytical fractal models for permeability and conductivity of open-cell metallic foams

It is demonstrated that analytical models of permeability and effective thermal conductivity for open-cell metallic foams can be developed using fractal theory, for the theory lends itself better to the problem at hand where the pore morphology of the foam is typically random. Upon determining the average tortuosity based on selected representative flow streamlines in a representative structure (RS) of the foam, the permeability as a function of porosity, average tortuosity and pore size ratio is analytically obtained, with no empirical or fitting parameter needed. Similarly, the analytical fractal model of effective thermal conductivity does not include any empirical or fitting parameter. Good agreement with experimental data verifies both models, squarely justifying the applicability of using fractal theory to characterize open-cell metallic foams. It is also demonstrated that the present fractal models can better characterize the randomness of pore distributions than previous permeability and conductivity models built upon simplified geometries including periodically distributed unit cells.

Thermophysical properties of porous Ti2AlC and Ti3SiC2 produced by powder metallurgy

The physicochemical properties of porous Ti2AlC and Ti3SiC2 MAX phase compounds with controlled porosity and grain size obtained by powder metallurgy techniques was studied in depth in order to access their suitability of applications such as catalytic devices on vehicles, heat exchangers or impact resistant structures. The study was performed on isostatic consolidated samples with different amount (20–60 vol%) and size of space holder (250–1000 µm) and in samples without space holder. Oxidation tests were performed at different temperatures for each material depending on their maximum service temperature. In order to understand the oxidation mechanism, oxidation kinetics were analysed to determine the influence of size and amount of porosity in each case. In addition, the microstructure and composition of the oxide layers formed after the tests were analysed by scanning electron microscopy (SEM). Electrical and thermal conductivity where studied at room temperature and at temperature up to 1000 °C. The effect of pore size and cell wall thickness is discussed. Permeability of foams was also measured. The effect of micro porosity and macro porosity on permeability is discussed. The coefficient of thermal expiation was also measured for all foams produced. It is established that these porous MAX phases have suitable properties for their use as catalytic substrates, heat exchanges, high temperature filters or volumetric solar receivers.

Permeability of Monodisperse Solid Foams

Fluid permeability of solid foams is a crucial parameter to control transport phenomena in numerous engineering applications, such as heat exchangers or filters for example. Open-cell foams with monodisperse pore diameter ranging from 200 to 1000 µm and solid volume fraction ranging from 0.1 to 0.38 are produced and Darcy permeability is measured. The permeability divided by the square of the pore size shows an exponential decay as a function of solid volume fraction. Surprisingly, existing models do not capture this exponential decay and actually they predict permeability values significantly larger than the measured values. The observed exponential decay is then successfully described by using models based on the viscous dissipation occurring through the apertures that connect the foam pores and by accounting for both the mean size of the apertures and the mean number of apertures per pore.

The importance of pore throats in controlling the permeability of magmatic foams

Vesiculation of hydrous melts at 1 atm was studied in situ by synchrotron X-ray tomographic microscopy at the TOMCAT beamline of the Swiss Light Source (Villigen, Switzerland). Water-undersaturated basaltic, andesitic, trachyandesitic, and dacitic glasses were synthesized at high pressures and then laser heated at 1 atm. on the beamline, causing vesiculation. The porosity, bubble number density, size distributions of bubbles, and pore throats, as well as their tortuosity and connectivity, were measured in three-dimensional tomographic reconstructions of sample volumes, which were also used for lattice Boltzmann simulations of viscous permeabilities. Connectivity of bubbles by pore throats varied from ~ 100 to 105 mm−3, and for each sample correlated with porosity and permeability. Consideration of the results of this and previous studies of the viscous permeabilities of aphyric and crystal-poor magmatic samples demonstrated that at similar porosities permeability can vary by orders of magnitude, even for similar compositions. Comparison of the permeability relationships from this study with previous models (Degruyter et al., Bull Vulcanol 72:63–74, 2010; Burgisser et al., Earth Planet Sci Lett 470:37–47, 2017) relating porosity, characteristic pore-throat diameters, and tortuosity demonstrated good agreement. Modifying the Burgisser et al. model by using the maximum pore-throat diameter, instead of the average diameter, as the characteristic diameter reproduced the lattice Boltzmann permeabilities to within 1 order of magnitude. Correlations between average bubble diameters and maximum pore-throat diameters, and between porosity and tortuosity, in our experiments produced relationships that allow application of the modified Burgisser et al. model to predict permeability based only upon the average bubble diameter and porosity. These experimental results are consistent with previous studies suggesting that increasing bubble growth rates result in decreasing permeability of equivalent porosity foams. This effect of growth rate substantially contributes to the multiple orders of magnitude variations in the permeabilities of vesicular magmas at similar porosities.

A Novel Method for Permeability Estimation from Micro-tomographic Images

A methodology has been developed to create a pore network model (PNM) from the geometrical/topological information extracted from the micro-tomographic images of a polyurethane (PU) foam sample. By solving fluid-flow equations on this model, we could estimate the permeability of PU foams and validate with experimental findings. Previous literature suggested that the permeability of open-cell PU foams was overestimated using the traditional Carman–Kozeny and Duplessis–Masliyah models. Inter-cell connectivity was deemed as the potential cause of this difference. Thus, taking into consideration the effects of spatial arrangement of pores of different sizes, the throat constriction between pores, and the percentage open-cell content, a PNM was employed where the entire void space is treated as a network of pores and throats of varying dimensions. To simulate flow through the network, the mass conservation equations were solved at each pore, the Darcy’s equation was applied globally, and thus, the overall permeability of the foam was estimated. Using our PNM, we were able to estimate the permeability with very good agreement with experimentally observed results. This indicates that pore connectivity has a significant effect on permeability. Parametric studies with PNM also revealed that the permeability depends on the square of pore size while is quite sensitive to throat constrictions with cubic dependence.

Gas permeability of Ti6Al4V foams prepared via gelcasting, experiments and modelling

The use of light components possessing tailored functional properties will represent one of the main issues for the development of new products. Open cells foams, where porosity is engineered in order to grant for lightness, high geometrical surface, interconnectivity and permeability, can be shaped using various techniques and possess regular or irregular structure. In this work, Ti6Al4V foams were produced using the gelcasting route, a technique usually employed for ceramics. Numerical simulations were carried out directly on 3D images of the foam’s microstructure obtained by X-ray 3D µCT in order to predict the gas permeability properties of the produced samples. The obtained results were compared to experimental measurements of gas permeability. This approach could be useful for the prediction of Newtonian fluids permeability behavior of non-periodic and complex porous structures.

Influence of flow conditions and foam parameters on pressure drop and heat transfer in flow of fluids through channels filled with metal foams

This paper reports the results of research into air, water and oil flow through pipes filled with open-cell metal foams, which form a material with a considerable potential for application in the design of process equipment. This study applied three metal foams with various geometrical parameters. The objective of the experiments involved the measurement of pressure drop within a relatively large range of the variability of the flow conditions. On the basis of data from the literature and analysis of the results of experiments, an assessment was undertaken concerned with the influence of the geometrical parameters of the foams as well as velocities and fluid properties on the flow regime and pressure drop. The results demonstrate that the theory concerned with the fluid flow through porous media has a limited application with regard to metal foams due to the considerable turbulence of the flow through such foams. If flow occurs in other conditions than laminar regime, the permeability of metal foams is relative not only to the geometrical structure of the foams but also depends on the properties and velocity of the fluid. The present study demonstrated that the assessment of the flow regime can apply the modified Reynolds number in which the characteristics dimension is defined on the basis of the parameters accounting for the geometrical foam structure. Three flow regimes were distinguished – laminar flow, transient Forchheimer and transient Froud flow. The ranges corresponding to the occurrence of the particular flow regimes were subsequently determined.

Pores Size Influence on the Permeability for Copper Foams

In the paper are presented the techniques for obtaining metallic foams based on copper. The permeability of them is described. The metal foams in this case are made from a mixture of copper and NaCl in well-established proportions (20%, 30% and 40% Cu), pressed (the mixture) in mould and sintered at temperatures of 780°C and 830°C. Proper knowledge’s of the permeability (K) are important and should be defined properly [1]. The present study was carried out to have an understanding the influence of different permeability values for different NaCl concentrations in the foams. The necessary equipment was adapted for this purpose. The permeability values obtained for this reason was correlated with the pore size and number of pores per square centimeter. With the increasing of pore diameters and copper proportion the increasing of permeability of metallic foams was also measured.

Tortuosity studies of cellular vitreous carbon foams

The tortuosity of cellular vitreous carbon (CVC) foams having different bulk densities and different pore sizes has been investigated by a diffusion technique. The corresponding values were compared to tortuosities determined by two other different methods: acoustic on one hand, and tomographic on the other hand. Whereas acoustic tortuosities were slightly higher than diffusional ones but followed the same linear increase with CVC density, geometrical ones were much lower and presented the opposite trend. Such apparent discrepancies were discussed and justified, and the concept of constrictivity was used for that purpose. As diffusional tortuosity is the relevant parameter for accounting for transport phenomena, the permeability of CVC foams could be calculated from such tortuosity values, based on a realistic pore model of cellular carbon foams. The agreement between calculated and measured permeabilities was quite satisfactory.

Erratum to: Permeability of aqueous foams

Erratum to: Permeability of aqueous foams

Permeability of porous gelcast scaffolds for bone tissue engineering

The permeability of metallic and ceramic open-cell foams prepared by the gelcasting technique was assessed by fitting of Forchheimer’s equation to experimental pressure drop curves. The ceramic composition was based on pure hydroxyapatite, while the metallic composition was based on titanium metal. Experimental Darcian (k 1) and non-Darcian (k 2) permeability constants displayed values in the range 0.40–3.24 × 10−9 m2 and 3.11–175.8 × 10−6 m respectively. Tortuosity was evaluated by gas diffusion experiments and ranged from 1.67 to 3.60, with porosity between 72 and 81% and average hydraulic pore size between 325 and 473 μm. Such features were compared to data reported in the literature for cancellous bones and synthetic scaffolds for bone graft. A detailed discussion concerning the limitations of Darcy’s law for fitting laboratory data and for predicting fluid flow through scaffolds in real biomedical applications is also performed. Pore size was obtained by image analysis and was also derived from permeation-absorption-diffusion experiments. In both cases, values were within the range expected for porous scaffolds applications.

Influence of sample thickness and measurement set-up on the experimental evaluation of permeability of metallic foams

The pressure drop measured when a fluid flow through a porous medium is presumed a linear function of the medium thickness along the flow direction. Recent works have, however, reported that such linear dependence is not valid for highly porous foams. The deviations observed were attributed to the relative contribution of entrance and exit effects on the total pressure drop. In this work, the effect of the measurement set-up and of the specimen thickness on the pressure drop was investigated. Permeability measurements were carried out with nickel-chromium foams. Two different set-ups for holding the samples were tested. Pressure drop vs. velocity curves were obtained with water at room temperature. The results obtained showed that the normalized pressure drop was affected by the specimen thickness and the set-ups used. The size of the annular section supporting the specimens has a clear influence on the magnitude of the deviation from the expected linear behavior between the pressure drop and the medium thickness. This effect has been attributed to the fluid diffusion and flow expansion into the annular region covered by the specimen support. The magnitude of this effect is more important as the thickness of the specimens increased. No significant entrance or exit effect was observed when the supporting annular section was minimized. The dispersion of permeability constants caused by this effect was determined and discussed.

Permeability of aqueous foams

We perform forced-drainage experiments in aqueous foams and compare the results with data available in the literature. We show that all the data can be accurately compared together if the dimensionless permeability of the foam is plotted as a function of liquid fraction. Using this set of coordinates highlights the fact that a large part of the published experimental results corresponds to relatively wet foams (epsilon approximately 0.1). Yet, most of the foam drainage models are based on geometrical considerations only valid for dry foams. We therefore discuss the range of validity of the different models in the literature and their comparison to experimental data. We propose extensions of these models considering the geometry of foam in the relatively wet-foam limit. We eventually show that if the foam geometry is correctly described, forced drainage experiments can be understood using a unique parameter --the Boussinesq number.

Specific Surface Area Model for Foam Permeability

Liquid foams were recognized early to be porous materials, as liquid flowed between the gas bubbles. Drainage theories have been established, and foam permeability has been modeled from the microscopic description of the equivalent pores geometry, emphasizing similarities with their solid counterparts. But to what extent can the theoretical work devoted to the permeability of solid porous materials be useful to liquid foams? In this article, the applicability of the Carman-Kozeny model on foam is investigated. We performed measurements of the permeability of foams with nonmobile surfactants, and we show that, in introducing an equivalent specific surface area for the foam, the model accurately describes the experimental data over two orders of magnitude for the foam liquid fraction, without any additional parameters. Finally, it is shown that this model includes the previous permeability models derived for foams in the dry foams limit.

Effective permeability and thermal conductivity of open-cell metallic foams via homogenization on a microstructure model

Abstract To predict effective thermal conductivities and permeabilities of open-cell foams, produced by a SlipReactionFoamSintering (SRFS)-process, a multiscale approach based on the homogenization method is applied. This formulation allows to calculate effective aerothermal properties by solving either special thermal or Stokes flow problems on a unit cell. Starting from a detailed foam description by tomographic images, 3D microstructure models of the SlipReaction (SR)-foam are generated by combining spectral analyses with a specific mesh generation. Effective thermal conductivities of an Inconel foam sample are predicted with different FE discretizations by the homogenization procedure and compared with the experimental results, measured by the transient plane source technique. Effective Darcy permeabilities are evaluated in the foam center and compared with experimental pressure drop measurements. Moreover, permeability predictions of the unit cells with extremal porosity allow to measure the permeability scatter of the Inconel SR-foam sample.

Characterization of the structure and permeability of titanium foams for spinal fusion devices

Titanium foams produced via the space-holder method are used for spinal fusion devices since their combination of an open-cell structure and bone-like mechanical properties promises potentially excellent bone ingrowth. Earlier studies have indicated that the size of the pores and interconnects must be greater than 100 μm for effective bone ingrowth and vascularization. Hence, the quantification of the pore and interconnect size is required for efficient scaffold design. In this study, microcomputed tomography (μCT) was used to obtain the three-dimensional (3D) structure of Ti foams with three levels of porosity (51%, 65% and 78%). Novel algorithms were then applied to quantify both the pore and interconnect size of Ti foams as a function of porosity. All foams possessed a modal pore and interconnect size in excess of 300 μm, satisfying the requirement of being greater than 100 μm. The pore and interconnect size also dominates the flow properties or permeability of open-cell structures. Therefore, the μCT data was also used to generate a mesh for computational fluid dynamics analysis to predict the permeability. The calculated permeability (117–163 × 10 −12 m 2 depending on direction) for the Ti foams with 65% porosity was first validated against experimental measurements (98–163 × 10 −12 m 2) and then compared to prior authors’ measurements in healthy cancellous bovine bone (233–465 × 10 −12 m 2). The close match among all the permeability values proves the suitability of the material for biomedical skeletal-implant applications.

Permeability and Thermal Transport in Compressed Open-Celled Foams

A computational methodology is proposed to describe the fluid transport in compressed open-celled metallic foams. Various unit-cell foam geometries are numerically deformed under uniaxial loads using a finite-element method. An algorithm is developed and implemented to deform the fluid domain mesh inside the unit-cell foam based on the deformed solid unit-cell geometry. Direct simulations of the fluid transport in these deformed meshes are then performed over a range of Reynolds numbers used in practical applications. The model is validated against available experimental results and correlations. A corrected model is proposed for the permeability of compressed foams as a function of strain for flows transverse to the direction of compression. The thermal conductivity of fluid-saturated foams is also computed. Compression of foams increases the conductivity transverse to the direction of compression and decreases the conductivity parallel to it.

Node contribution to the permeability of liquid foams

This paper deals with the drainage of liquid foams. The liquid velocity is known to be related to viscous dissipation occurring within the elements of the liquid network, i.e. the channels and the nodes. When compared together, available values for the hydrodynamic resistance of a foam node appear to span over more than one order of magnitude. To clarify this point, we propose an alternative experimental method to estimate the value of this parameter. In contrast to previous experimental work performed on the foam scale, the node resistance is not treated as a fitting parameter, but instead it is measured directly on the microscopic scale. The results allow a consistent range of values to emerge for this parameter.