REPORTS A N D COMMENTS
SYMPOSIUM ON FLOW THROUGH POROUS MEDIA The sixth summer state-of-the-art symposium to be sponsored by the I&EC Division of the ACS will be held at the Carnegie Institution in Washington, D.C., on June 9-11, 1969. Abstracts of the papers which will be presented at the three-day symposium are presented here. Attendees are encouraged to preregister and a preregistration form for this purpose is to be found below. I t will also be possible to register at the meeting site but a special preregistration package price makes advance registration financially attractive.
THE CHAIRMAN Dr. Richard J. Nunge, chairman of the I&EC Division Symposium on Flow Through Porous Media, is Associate Professor of Chemical Engineering at Clarkson College of Technology, Potsdam, N.Y. 13676. He was Assistant Professor at Clarkson from 1965 to 1968 and before that was Instructor in Chemical Engineering at Syracuse University from 1962 to 1964. Dr. Nunge obtained his B.Ch.E. at Syracuse University in 1960, his M.Ch.E. from Syracuse in 1962, and his Ph.D. from Clarkson College of Technology in 1965. Professor Nunge has published a number of papers in the areas of gas-liquid kinetics, multistream heat transfer, and dispersion. He is currently one of the authors of I&EC’s Annual Review of Fluid Dynamics. Dr. Nunge is a member of the American Institute of Chemical Engineers, the Institute of Colloid and Surface Chemistry, Sigma Xi, and AAUP.
Flow Field in a Bed of Spheres at High Reynolds Numbers. T. J. Hanratty and A. J. Karabelas, University of Illinois, Urbana. Transport phenomena in beds of spheres at high Reynolds numbers will be briefly reviewed. Particular emphasis will be placed on those studies which could be of some help in formulating a model for the flow field. Most researches on flow through packed beds reflect the average behavior of a large number of particles. I n recent years, results have been obtained which give more detailed information. These include studies of the motion of dye streamers and of the variation of the pressure, the local mass transfer rate, and the local shear stress around one of the spheres in a packed bed of spheres.
O n the Flow of Suspensions Through Porous Media. J. P. Herzig and P. Le Goff, Universiti de Nancy, France Because of its use in the treatment of polluted waters, the flow of suspensions through porous media has often been studied. This phenomenon is very complex owing to the diversity of the mechanisms involved. I t is described mathematically by the continuity equations of the fluid and the particles, the kinetic equation of clogging, and the connection between the retention of particles and the pressure drop of the fluid. Generally, the continuity equations may be simplified by neglecting the diffusional term, and also the concentration of suspended particles with respect to the concentration of retained particles. The kinetic equation is assumed to be of first order with respect to the concentration of particles in the suspension, as the experiments show. T h e “dec1ogging”--i.e., the breakaway of already deposited particles, is often neglected.
From these two equations, a new differential equation between the most important parameters is demonstrated, from which the author deducts general macroscopic properties independent of the experimental system considered. These properties present a formal analogy with adsorption phenomena on columns (chromatography). However, the relation between the experimentally determined macroscopic parameters and the elementary mechanisms of clogging remains to be established accurately. T o achieve this, the elementary mechanisms themselvesLe., retention sites, retention forces, the capture and breakaway processes, need to be better known. I n addition, the influence of the retained particles on the permeability of the clogged bed is not fully established.
Diffusion and Flow of Gases in Porous Solids. G. R. Youngquist, Clarkson College of Technology, Potsdam, N. Y. Methods for predicting and correlating the diffusion and flow of gases in porous solids such as catalysts and adsorbents are reviewed. T h e problem is very complex. T h e pore geometry of these solids is not well understood, and frequently several transport mechanism, such as bulk diffusion, Knudsen diffusion, and viscous and slip flow contribute to the flux of gas through the pores. T h e contribution of each of these mechanisms varies with pore size and gas pressure. Most models for prediction of gaseous diffusion and flow in porous systems are based on reasonably well developed theories for diffusion and flow in capillaries. An exception is the “dusty gas” model in which the particles composing the porous medium are treated as large stationary molecules forming a component of the gas mixture.
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At low pressures and small pores, where the mean free path of the diffusing molecules is much larger than the capillary diameter, Knudsen diffusion predominates. For high pressures and/or large pores, ordinary bulk diffusion is dominant, while for intermediate ranges both contributions may be important. I n the presence of a pressure gradient, viscous and slip flow may also contribute to the flux. The capillary theories developed for these regimes have been adapted for use with porous solids in various ways, dependlng on the pore size distribution of the solid. For solids with narrow pore size distribution, simple parallel pore models have been used with a mean effective pore size and equivalent parallel pores. For solids with bidisperse pore size distributions, random pore models which account for parallel and series diffusion in micro- and macropores have been employed. Most models of these types contain a n adjustable parameter such as the tortuosity which must be determined from experiment. Reasonably good agreement of theory with experiment has been obtained in some cases, but no general method for prediction of fluxes is yet available.
Theoretical Framework for Coalescence by Porous Media. L. A. Spielman, Harvard University, Cambridge, and S. L. Goren, University of California, Berkeley Flow through porous solids provides the basis for a number of phase separation operations, among these being the coalescence of finely dispersed liquid suspensions. A theoretical framework for coalescence by porous media is given through a set of coupled equations in which concepts from air and water filtration describe the local capture of suspended droplets within the medium, and two-phase flow theory, developed largely in soil physics and petroleum reservoir engineering, describes the flow of coalesced liquid through the matrix. Within the framework, the filter coefficient, capillary pressure, and fluid permeabilities appear explicitly, but their detailed mathematical forms are required to complete the scheme. T o this end, appropriate literature from the respective areas of particulate filtration and multiphase flow is reviewed.
Advances in the Theory of Fluid Motion i n Porous Media. Stephen Whitaker, University of California, Davis Recent theoretical studies of flow in porous media have been reviewed. Based on these works, a general theory of creep flow in a rigid porous medium is proposed.
Statistical Models of Flow Through Porous Media. K. H. Liao and A. E. Scheidegger, University of Illinois, Urbana The paper reviews the models advocated in the literature for the description of flow through porous media. Special emphasis is placed on statistical models which aim at a theoretical explanation of the occurrence of dispersive effects. These models are a random-walk model, and new random-topology models are presented here for the first time.
Flow of NowNewtonian Fluids Through Porous Media. J. G . Savins, Mobil Research and Development Corp., Paulsboro, N. J. T h e contemporary literature relates interesting descriptions of how rheologically complex fluids behave in flow through a variety of porous media. From these studies, several correlations of flow phenomena with the rheology of grossly 8
“non-Newtonian” and “Newtonian-like” fluids have been described or attempted. I n the present paper we will be mainly concerned with the phenomenological aspects of these effects. These effects occur under different flow conditions in porous structures and seem to be coupled with the unusual rheological behavior of complex fluids. We will begin with a necessarily restricted overview of non-Newtonian fluids, focusing our attention on those rheological characteristics which appear to describe, at least under conditions of viscometric flow, the fluid systems of contemporary interest in studies of non-Newtonian flow through porous media. Phenomena likely to have a role in mechanisms responsible for the behavior observed in non-Newtonian flow through porous media’will also be recognized. Some of these non-Newtonian effects have long been recognized and treated as the principal modes of expression of “non-Newtonian behavior.’’ Other effects have not been considered as such in the past. Ironically, as chemically distinct non-Newtonian systems are investigated, these equally important and different manifestations of complex flow behavior have been “red i s c o v e r e d , ” ~to ~ speak. We will address some brief remarks concerning the diverse kinds of porous media which have been used in experiments involving complex fluids. No attempt will be made to discuss the structure and properties of porous materials per ze or to consider topics related to such items as pore size distribution and local voidage characteristics. These topics and discussions of advances in this area are thoroughly reviewed elsewhere in this symposium. A brief summary of the characteristics of known fluid systems relating to non-Newtonian flow through porous media will be presented. We will deal with the generally accepted techniques for reduction and analysis of data from flow of non-Newtonian fluids in porous media. We consider, next, examples out of the literature which illustrate the fascinating, and sometimes bizarre flow behavior of complex fluids in different models of porous media. Different descriptions are found in the literature. Several mechanisms explaining the different phenomena will be illustrated and discussed. Finally we consider some applications to show the relevance of non-Newtonian flow through porous media to processes of technological importance. Applications in such diverse areas as oil recovery, lubrication, and naval architecture will be noted, and examples from the patent art will be cited.
Dispersion in Heterogeneous, Nonuniform, Anisotropic Porous Media. R. A. Greenkorn and D. P. Kessler, Purdue University, Lafayette, Ind. T h e phenomenon of dispersion during flow in porous media has received increasing attention in the past 15 years. Much of the literature until recently was concerned with solutions to the diffusion equation adapted for dispersion and with measurements interpreted by fitting such solutions (for specified boundary conditions) to experimental data. Although the effects of particle size distribution and heterogeneity have been considered, this has been done only in a limited number of studies and then by semiempirical correction factors. I n the past few years there has been concern with the mechanisms that cause dispersion; statistical models which attempt to examine nonuniformity and anisotropy; continuum theories for general, nonuniform, anisotropic porous media; and some measurements concerned with the effects of nonhomogeneities. This review examines the mechanisms of dispersion and defines the nonhomogeneities hetero-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
geneity, nonuniformity, and anisotropy. Statistical models are considered to see how the properties of the media affect dispersion. The continuum theories are revicwed and the limiting assumptions required to use the usual diffusion equation to interpret dispersion data are discussed. Experimental data are reviewed in light of the statistical and continuum theories and the scale of the system and its elements are considered. Finally, the use of a parametric form of a dispersion coefficient to include the effects of viscosity, density, and media stratification is discussed.
Reverse Osmosis. S. Sourirajan and J. P. Agrawal, National Research Council of Canada, Ottawa The reverse osmosis process is discussed with particular reference to systems involving aqueous solutions and Loeb-Sourirajan-type porous cellulose acetate membranes. Mechanisms of the process and porous cellulose acetate membrane technology are briefly reviewed. Based on a general capillary diffusion model for the transport of solvent and solute through the membrane, transport equations applicable for the entire range of solute separation are presented. The results of the analysis and correlations of the experimental reverse osmosis data are illustrated. O n the basis of the above equations and correlations, methods of membrane specification, expressing membrane selectivity and predicting membrane performance are outlined. Reverse osmosis is then treated as a unit operation in chemical engineering. A set of general equations for reverse osmosis process design is then derived for reverse osmosis systems specifirtd in terms of membrane specifications and operating conditions. The utility of these equations for studying system performance and process parameters is then illustrated. Reverse osmosis as a general concentration process and the separation of mixed solutes in aqueous solution are then briefly discussed from the points of view of process design and the predictability of membrane performance, respectively.
Anisotropy in Porous Media. D. Fontugne, A. Barduhn, and P. Rice, Syracuse University Most studies on directional permeabilities in the past have dealt mainly with consolidated sand and were aimed at developing more efficient recovery of oil from producing sands. Recently, however, interest rose in directional permeabilities in unconsolidated media, particularly ice or hydrate crystals from desalination plants using a freezing or hydrate process. Permeability data on both consolidated and unconsolidated beds are reviewed here. I n addition, the effects of the bed formation processes and methods of measurement are reviewed. Finally, methods of predicting multidimensional permeabilities from the particle shape bed porosity are discussed. Abundant permeability data are available on consolidated sands. Ordinarily the permeabilities parallel to the bedding plane are greater than those normal to it with a ratio ranging from 1 to as high as 42. I n a few cases, however, the permeability normal to the bedding plane exceeds that parallel to it. Previous permeabilities studies have also been conducted as a function of the angular direction in the bedding plane. These measurements show a remarkable consistency in the directional flow trend throughout a given sand formation. T h e permeability parallel to the bedding plane may vary 25 to 30% depending on the angular direction.
The measurements in this laboratory on unconsolidated beds show that for the natural settling of irregular particles, the vertical permeabilities are less than horizontal permeabilities with a difference of the order of 20%. Anisotropy in unconsolidated media decreases as the porous bed settles down. During the formation of a bed, a body sinking in a liquid tends to sink with its longest dimension horizontal. This has been observed experimentally for a single particle falling in a liquid of infinite extent. This implies a preferred horizontal orientation in porous beds formed in this manner. This process followed by consolidation might occur also in the formation of consolidated sand. Other factors such as ocean currents may also be influential in geological formations. The Carman-Kozeny equation has related poorly the vertical permeability data to the porosity of an anisotropic media. I n any event, this equation cannot predict the degree of anisotropy unless some generalized equivalent diameter of the particles can be found which would take the direction into account. Presently there is no adequate procedure for defining such a generalized diameter. Further studies on unconsolidated beds are in progress. An apparatus has been designed which allows the simultaneous measurement of vertical and horizontal permeab sional flow equations in anisotropic porous media can be solved both analytically for simple enough geometries and numerically in the more complex case of our apparatus. Complete networks of equipotential lines and streamlines were found for a given degree of anisotropy. This numerical study together with the experimental result allows for the determination of the degree of anisotropy. Numerical and analytical solutions can be found also for three-dimensional flow, but with a commensurate increase in difficulty and/or computational time.
Dispersion and Miscible Displacements in Porous Media. R. J. Nunge and W. N. Gill, Clarkson College of Technology, Potsdam, N. Y. T h e development of a predictive theory to describe accurately the mechanisms of mass transfer for miscible displacements taking place in porous media or the dispersion of solute in a flowing stream in a porous media is hampered by the complex and highly irregular structure of the media. One widely used technique for correlating experimental data has been a onedimensional diffusion equation containing an effective axial diffusion coefficient determined by fitting experimental data. The difficulty in using this type of macroscopic dispersion model lies in extrapolation to new situations, since it contains little information concerning the mechanisms of dispersion. One line of investigation, aimed at improving the interpretation of experimental measurements and delineating the mechanisms of dispersion likely to be important in a given circumstance, has been developed by viewing porous media as consisting of a microstructure, the interstices between particles, in which the microscopic laws of convective-diffusion apply. Since the physical laws which describe mass transfer are the same in every environment, and since these laws are linear, it is possible to study dispersion in various simple geometries wherein one or more of the effects present in a porous media are accounted for. The results can then be superimposed to obtain a mechanistic description of
dispersion in a porous media. Because of the complex structure of porous media, a number of additional effects not easily included in simple models are to be expected and hence the ultimate goal has not yet been achieved. However, many interesting and important conclusions related to the application of macrostructure laws such as the one dimensional diffusion model have been reached. This paper reviews the progress which has been made in this area.
Pore Structure Analysis. F. A. L. Dullien, University of Waterloo, Canada An attempt has been made systematically to survey the literature from 1953 to 1969 dealing, mainly, with the individual methods of investigating pore structures. A few papers dating before 1953 have been also included. Greatest activity has been evidenced in the field of adsorption methods, and mercury porosimetry. I n both fields of investigation, interesting new theories have been put forward permitting improved interpretation of the experimental data. Development of a variety of automatic and computerized equipment for pore size distribution determination has been reported. Among a variety of other methods of pore structure investigation, considerable attention has been paid also to the relationship between pore structure, on the one hand, and capillarity, displacement of oil by water, permeability, and diffusion, and flow of gases, on the other. Radiographic, microscopic, and electron microscopic pore structure studies also have received some attention. Only a few papers discussed theoretical problems of pore structure analysis in general, without reference to any particular experimental method of investigation. A number of important papers presented integrated analyses of the pore structure of certain types of materials such as carbons, graphite, porous electrodes, filter media, concrete, ceramic bricks, paper, textiles, and wood.
Physics and Thermodynamics of Capillary Action in Porous Media. N. R. Morrow, Esso Production Research Co., Houston, Tex. A thermodynamic account of immiscible displacement in porous media is developed in which work of displacement is related to changes of surface area in the system. Displacement is inherently irreversible, but if conducted infinitely slowly, it is shown to be quantized as an alternate series of reversible displacements (named isons) each of which is terminated by an unstable fluid configuration resulting in spontaneous redistribution of fluid at constant saturation (named rheon). I n fine detail, curves relating capillary pressure to pore space saturation are not smoothly continuous; but, the area under a capillary pressure curve is equal to the reversible external work done on or by the system and is precisely equal to the area under its isons. Each rheon is characterized by a definite net decrease in free energy; for given displacements, there are inherent thermodynamic efficiencies of exchange between surfacefree energy and external work. Experiments are described through which efficiencies for drainage and imbibition in random packings of spheres are determined.
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