Condensation in porous media

Tuller, M., D. Or, and L.M. Dudley. 1999. Adsorption and capillary condensation in porous media Liquid retention and interfacial configurations in angular pores. Water Resour. Res. 35 1949-1964. Tuller, M., and D. Or, 2001, Hydraulic conductivity of variably saturated porous media Film and corner flow in angular pore space. Water Resour. Res. (In press.) van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44 892-898. 

For plain media, the film evaporation adjacent to a heated vertical surface is similar to the film condensation. In porous media, we also expect some similarity between these two processes. For the reasons given in the last section, we do not discuss the cases where 8gld — 1, where 8g is the vapor-film region thickness. When 8gld 1 and because the liquid flows (due to capillarity) toward the surface located at y = 8g, we also expect a large two-phase region, that is, ld 1. Then a local volume-averaged treatment can be applied. 

In porous media, liquid-gas phase equilibrium depends upon the nature of the adsorbate and adsorbent, gas pressure and temperature [24]. Overlapping attractive potentials of the pore walls readily overcome the translational energy of the adsorbate, leading to enhanced adsorption of gas molecules at low pressures. In addition, condensation of gas in very small pores may occur at a lower pressure than that normally required on a plane surface, as expressed by the Kelvin equation, which relates the radius of a curved surface to the equilibrium vapor pressure [25],... 

Only a few studies have been published, showing capillary condensation. Although separation by capillary condensation is not new at all, but has been widely used in separation processes exploiting porous adsorbents, the dynamic behavior of flow of capillary condensate through porous media has received little attention. And it is this dynamic behavior that is important when capillary condensation is used as a separation mechanism. 

FIGURE 5.27 Transport mechanisms in porous media molecular or gaseous flow, (a) Knudsen flow, (b) surface diffusion, (c) multilayer diffusion, (d) capillary condensation, and (e) configurational diffusion. 

The book provides a comprehensive coverage of the subject giving a full discussion of forced, natural, and mixed convection including some discussion of turbulent natural and mixed convection. A comprehensive discussion of convective heat transfer in porous media flows and of condensation heat transfer is also provided. The book contains a large number of worked examples that illustrate the use of the derived results. All chapters in the book also contain an extensive set of problems. 

Figure 4. Schematic representation of transport mechanisms in porous media (a) Poiseuille flow (b) Knudscn diffusion (c) surface diffusion (d) capillary condensation (e) molecular sieving...

The question of the constancy of the surface tension in porous media has been under consideration for many years and has been taken up again recently by Grown et al. (1997). Formerly, it was thought that for a concave liquid-vapour interface the surface tension should increase with increased curvature. The experimental findings that the hysteresis critical temperature is generally appreciably lower that the bulk critical temperature (see Section 7.5) is considered to be a strong indication that the surface tension of a capillary-condensate is reduced below the bulk value. More work on model pore structures is evidently required to settle this question. 

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

Many numerical models make additional assumptions, valid if only some specific questions are being asked. For example, if one is not interested in the start-up phase or in changing the operation of a fuel cell, one may apply the steady state condition that time-independent solutions are requested. In certain problems, one may disregard temperature variations, and in the free gas ducts, laminar flow may be imposed. The diffusion in porous media is often approximated by an assumption of isotropy for the gas diffusion or membrane layer, and the coupling to chemical reactions is often simplified or omitted. Water evaporation and condensation, on the other hand, are often a key determinant for the behaviour of a fuel cell and thus have to be modelled at some level. 

Jaguste D.N. and Bhatia S.K., Combined surface and viscous flow of condensable vapor in porous media, Chem. Eng. Sci. 50 167 (1995). 

Udell K.S. (1983) Heat transfer in porous media heat from above with evaporation, condensation and capillary effects, J. Heat Transfer, vol. 105, pp.485-492. 

The case of combined buoyant-forced (i.e., applied external-pressure gradient film condensation flow in porous media) has been examined by Renken et al. [112], and here we examine the case when there is no external pressure gradient. 

In the following, we examine the experimental results [127], that support these two asymptotic behaviors, that is. Bo > 1, the high-permeability asymptote, and Bo 1, the low-permeability asymptote. Then, we discuss the low-permeability asymptote using a onedimensional model [109,125,128], The one-dimensional model is also capable of predicting qa, that is, the onset of dryout. We note that the hysteresis observed in isothermal two-phase flow in porous media is also found in evaporation-condensation and that the q versus T0 - Ts curve shows a decreasing q (or T0 - Ts) and an increasing q (or T0 - Ts) branch. 

K. S, Udell, and J. S. Fitch, Heat and Mass Transfer in Capillary Porous Media Considering Evaporation, Condensation and Noncondensible Gas Effects, in Heat Transfer in Porous Media and Particulate Flows, ASME HTD, (46) 103-110,1985. 

In this chapter, the effect of capillary condensation upon catalytic reactions in porous media has been reviewed. It was shown that capillary condensation could have a strong influence upon catalytic reactions on its kinetics, transient dynamics, and catalyst pellet effectiveness factor. The reaction rate in the liquid phase is usually slower than in the gas phase due to the difference in adsorption equilibrium, and due to low solubility of hydrogen in the liquid (in hydrotreatment processes). 

Rajniak, P. and Yang, R.T., Unified network model for diffusion of condensable vapours in porous media, AIChEJ., 42, 319-331, 1996. 

Heat and mass transfers in porous media are coupled in a complicated way. On the one hand, heat is transported by conduction, convection, and radiation. On the other hand, water moves under the action of gravity and pressure gradient whilst the vapor phase moves by diffusion caused by a gradient of vapor density. Thus, the heat transfer process can be coupled with mass transfer processes with phase changes such as moisture sorption/desorption and evaporation/condensation. 

Effect of curvature on saturation pressure condensation and vaporization in porous media. 

Permporometry is the only method suitable for the determination of the size distribution of the active pores with diameters ranging from about 1.5 nm to 0.1 pm in porous media, particularly those with an asymmetric (and/or composite) structure. Permporometry, a relatively new technique, is based on the controlled blocking of the pores by capillary condensation and simultaneous measurement of the gas diffusional flux through the remaining open pores [86]. There are two different approaches of the method. 

Adsorption equilibria are normally considered in connection with processes occurring in porous media filters, catalysts, adsorbents, chromatographs, and rock of petroleum reservoirs. In macroporous and mesoporous media, adsorption is normally accompanied by another surface phenomenon, the capillary condensation. These two types of surface phenomena are closely connected because they are both produced by surface forces. On the other hand, these phenomena are relatively independent and may, to some extent, be discussed separately [3]. Moreover, the description of the coexistence of the adsorbed films and capillary condensate in the same capillary is a nontrivial problem. We present capillary condensation and adsorption separately, although their eommon roots are discussed in Section II. The (more or less) comprehensive description of the thermodynamics of multicomponent capillary condensation... 

The problem of capillary equilibrium in porous media is complicated from both experimental and theoretical points of view. The mechanisms of saturation and depletion of the porous medium are essentially nonequilibriiun. Further equilibration is due to slow processes like diffusion. The process of equilibration may be unfinished, since no significant changes of fluid distribution may occur during the time of an experiment. This especially relates to the so-called discontinuous condensate existing in the form of separate drops. As a result, thermodynamic states, which are not fully equilibrated, are interpreted from the practical point of view as equilibrium [28]. To the best of our knowledge, a consistent theory of such quasiequihbrium states has not yet been developed. In the following, we discuss the possible states of the two-phase mixtures in a porous medium, assuming complete thermodynamic equihbrium. This serves as a first approximation to a more complicated picture of the realistic fluid distribution in porous media. 

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