Media, porous

FIRE SPREAD ON SURFACES AND THROUGH SOLID MEDIA 

6 Flow through packed beds of particles (porous media) 

The flow of non-Newtonian liquids through beds of particles is treated in an analogous way to that adopted in Chapter 3 for the flow through ducts of regular cross-section. No complete analytical solution is, however, possible and a degree of empiricism complemented by the use of experimental results is often necessary. Firstly, however, the basic nature and structure of porous media (or beds of particles) will be briefly discussed. 

Of the numerous macroscopic parameters used to quantify porous media, those gaiiung widest acceptance in the hterature for describing the flow of single phase fluids are voidage, speciflc surface, permeability and tortuosity. Their values can often be inferred from experiments on the streamline flow of single phase Newtonian fluids. 

In addition, the speciflc surface, Sb, of the bed affects both its general structure and the resistance it offers to flow. It is defined as the surface area per unit volume of the bed, i.e. n ln . Hence, 5g can be expressed in terms of the voidage e and the specific area 5 of the particles 

For a given shape, S is inversely proportional to the particle size. Highly porous fibre glasses have specific surface areas in the range 5-7 x 10 m /m while compact limestones (s 0.04-0.10) have specific smface areas in the range 0.2-2 x lO mVm.  

Organic molecules spontaneously form corresponding cation-radicals on inclusion within activated zeolites (Yoon and Kochi 1988, Yoon 1993, Pitchumani et al. 1997). Zeolites are crystalline alu-mosilicate minerals that are widely used as sorbents, ion exchangers, catalysts, and catalyst supports. As zeolites act as electron acceptors due to the presence of Lewis- or Broensted-acid sites, confined organic compounds occur to be electron donors. Frequently, the interaction of electron donor with electron acceptor centers spontaneously generates cation-radicals and traps the ejected electrons. 

Let us compare M-ZSM-5 zeolites with M = H+, Li+, Na, K+, Rb, Cs, AF+, on one hand, and organic electron donors of variable ionization potentials, on the other. Zeolite H-ZSM-5 generates cation-radicals from substrates with an oxidation potential of up to 1.65 V (Ramamurthy et al. 1991). The naphthalene sorption by Al-ZSM-5 zeolites calcified in an atmosphere of oxygen or argon leads to the appearance of two occluded particles—the naphthalene cation-radical and isolated electron. Both particles were fixed by ESR method. Back reaction between the oppositely charged particles proceeds in an extremely slow manner and both the signals persist over several weeks at room temperature (Moissette et al. 2003). 

Easily ionizable anthracene forms the cation-radical as a result of sorption within Li-ZSM-5. In case of other alkali cations, anthracene was sorbed within M-ZSM-5 as an intact molecule without ionization (Marquis et al. 2005). Among the counterbalancing alkali cations, only Li+ can induce sufficient polarization energy to initiate spontaneous ionization during the anthracene sorption. The lithium cation has the smallest ion radius and its distance to the oxygen net is the shortest. The ejected electron appears to be delocalized in a restricted space around Li+ ion and Al and Si atoms in the zeolite framework. The anthracene cation-radical appears to be in proximity to the space where the electron is delocalized. This opens a possibility for the anthracene cation-radical to be stabilized by the electron s negative field. In other words, a special driving force for one-electron transfer is formed, in case of Li-ZSM-5. 

Cation-radicals, stabilized in zeolites, are excellent one-electron oxidizers for alkenes. In this bimolecular reaction, only those oxidizable alkenes can give rise to cation-radicals, which are able to penetrate into the zeolite channels. From two dienes, 2,4-hexadiene and cyclooctadiene, only the linear one (with the cylindrical width of 0.44 nm) can reach the biphenyl cation-radical or encounter it in the channel (if the cation-radical migrates from its site toward the donor). The eight-membered ring is too large to penetrate into the Na-ZSM-5 channels. The cyclooctadiene can be confined if the cylindrical width is 0.61 nm, however the width of the channels in Na-ZSM-5 is only 0.55 nm. No cyclooctadiene reaction with the confined biphenyl cation-radical was detected despite the fact that, in solution, one-electron exchange between cyclooctadiene and (biphenyl) proceeds readily (Morkin et al. 2003). 

The restriction for a nucleophile to penetrate and react with the confined cation-radical sometimes leads to unexpected results. Comparing the reactions of thianthrene cation-radicals, Ran-gappa and Shine (2006) refer to the zeolite situation. When thianthrene is absorbed by zeolites, either by thermal evaporation or from solution, thianthrene cation-radical is formed. The adsorbed cation-radical is stable in zeolite for a very long time. If isooctane (2,2,4-trimethylpentane) was used as a solvent, tert-butylthianthrene was formed in high yield. The authors noted it is apparent that the solvent underwent rupture, but the mechanism of the reaction remains unsolved.  

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The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

Capillary pressure gradients and Marongoni flow induce flow in porous media comprising glass beads or sand particles [40-42], Wetting and spreading processes are an important consideration in the development of inkjet inks and paper or transparency media [43] see the article by Marmur [44] for analysis of capillary penetration in this context. 

An important application of foams arises in foam displacement, another means to aid enhanced oil recovery. The effectiveness of various foams in displacing oil from porous media has been studied by Shah and co-workers [237, 238]. The displacement efficiency depends on numerous physicochemical variables such as surfactant chain length and temperature with the surface properties of the foaming solution being an important determinant of performance. 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

The success of transport models must be measured by their ability to describe the results of flow and diffusion measurements in porous media. 

When a pure gas flows through a channel the accompanying fall in pressure is accounted for partly by acceleration of the flowing stream and partly by momentum transfer to the stationary walls. Since a porous medium may be regarded as an assembly of channels, similar considerations apply to flow through porous media, but in the diffusional situations of principal interest here accelerational pressure loss can usually be neglected. If more than one molecular species is present, we are also interested in the relative motions of the different species, so momentum transfers by collisions between different types of molecules are also important. 

D Arcy s work was published in a book with the unlikely title "Les fontanes publiques de la ville de Dijon." The porous media in question were filter beds through which the water for the fountains circulated. 

The relation between the dusty gas model and the physical structure of a real porous medium is rather obscure. Since the dusty gas model does not even contain any explicit representation of the void fraction, it certainly cannot be adjusted to reflect features of the pore size distributions of different porous media. For example, porous catalysts often show a strongly bimodal pore size distribution, and their flux relations might be expected to reflect this, but the dusty gas model can respond only to changes in the... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

Chapter 6. SOME IMPORTANT EXPERIMENTAL RESULTS ON GAS MOTION IN POROUS MEDIA And capillaries... 

Proposed flux models for porous media invariably contain adjustable parameters whose values must be determined from suitably designed flow or diffusion measurements, and further measurements may be made to test the relative success of different models. This may involve extensive programs of experimentation, and the planning and interpretation of such work forms the topic of Chapter 10, However, there is in addition a relatively small number of experiments of historic importance which establish certain general features of flow and diffusion in porous media. These provide criteria which must be satisfied by any proposed flux model and are therefore of central importance in Che subject. They may be grouped into three classes. 

Thus his experiments were the first to indicate the surprising result that relation (6,1) remains valid even in conditions where bulk diffusion resistance is completely dominant. Accordingly (6.1), perhaps the most important single experimental result on diffusion in porous media, will be referred to as Graham s relation. 

Though by no means a complete theory, this is at least a reasonable explanation of the Knudsen minimum, and it then remains to explain why the minimum is not observed for flow through porous media. Pollard and Present attributed this to the limited length/diameter ratio of the channels in a typical porous medium and gave a plausible argument in favor of this view. 

In a later, but less detailed analysis of flow in the intermediate pressure range, Hiby and Pahl [37] suggested that the minimum should be absent when the length/diameter ratio of a capillary is less than about sixteen. Since it is likely that this is the case for the channels in most porous media of... 

Chapter 8. MODELS OF FLOW AND DIFFUSION IN POROUS MEDIA... 

Having discussed at some length the formulation and testing of flux models for porous media, we will now review v at Is, perhaps, their most Important application - the formulation of material balances In porous catalyst pellets. 

C. Carman, Flow of Gases -Through Porous Media, Academic Press,... 

Humidification. For wiater operation, or for special process requirements, humidification maybe required (see Simultaneous HEAT and mass transfer). Humidification can be effected by an air washer which employs direct water sprays (see Evaporation). Regulation is maintained by cycling the water sprays or by temperature control of the air or water. Where a large humidification capacity is required, an ejector which direcdy mixes air and water in a no22le may be employed. Steam may be used to power the no22le. Live low pressure steam can also be released directly into the air stream. Capillary-type humidifiers employ wetted porous media to provide extended air and water contact. Pan-type humidifiers are employed where the required capacity is small. A water filled pan is located on one side of the air duct. The water is heated electrically or by steam. The use of steam, however, necessitates additional boiler feed water treatment and may add odors to the air stream. Direct use of steam for humidification also requires careful attention to indoor air quahty. 

The framework for the solution of porous media flow problems was estabUshed by the experiments of Henri Darcy in the 1800s. The relationship between fluid volumetric flow rate, hydraulic gradient, and cross-sectional area, yi, of flow is given by the Darcy formula ... 

The WAG process has been used extensively in the field, particularly in supercritical CO2 injection, with considerable success (22,157,158). However, a method to further reduce the viscosity of injected gas or supercritical fluid is desired. One means of increasing the viscosity of CO2 is through the use of supercritical C02-soluble polymers and other additives (159). The use of surfactants to form low mobihty foams or supercritical CO2 dispersions within the formation has received more attention (160—162). Foam has also been used to reduce mobihty of hydrocarbon gases and nitrogen. The behavior of foam in porous media has been the subject of extensive study (4). X-ray computerized tomographic analysis of core floods indicate that addition of 500 ppm of an alcohol ethoxyglycerylsulfonate increased volumetric sweep efficiency substantially over that obtained in a WAG process (156). 

Sulfonates for Enhanced Oil Recovery. The use of hydrocarbon sulfonates for reducing the capillary forces in porous media containing cmde oil and water phases was known as far back as 1927—1931 (164,165). Interfacial tensions between 10 and 10 N/m or less were estabUshed as necessary for the mobilization and recovery of cmde oil (166—169). 

Viscosity is an important property of calcium chloride solutions in terms of engineering design and in appHcation of such solutions to flow through porous media. Data and equations for estimating viscosities of calcium chloride solutions over the temperature range of 20—50°C are available (4). For example, at 25°C and in the concentration range from 0.27 to 5.1 molal (2.87—36.1 wt %) CaCl2, the viscosity increases from 0.96 to 5.10 mPa-s (=cP). 

P. C. Carman, Floiv of Gases through Porous Media, Butterworths Pubheations Ltd., London, 1956. 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

Creeping flow (Re <- 1) through porous media is often described in terms or the permeability k and Darcy s Law ... 

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation ... 

For gas flow through porous media with small pore diameters, the slip flow and molecular flow equations previously given (see the Vacuum Flow subsec tion) may be applied when the pore is of the same or smaller order as the mean free path, as described by Monet and Vermeulen (Chem. E/ig. Pi og., 55, Symp. Sei , 25 [1959]). 

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