I n d . Eng. Chem. Res. 1988, 27, 1936-1941
1936
V, = volume of the reactor (6.36 X lo-* m3) W , = mass flow rate of water, kg/s AP = pressure difference (measured) between points 10 and 11 in Figure 1, mHzO APf = pressure drop on the reactor due to friction between points 10 and 11 in Figure 1, mHzO Up, AP, = pressure drop on the reactor due to friction between points 10 and ll in Figure l in the presence of particles and in the absence of particles, respectively Greek Symbols 7 = defined in eq 6 p, = viscosity of water, kg/ms pp, pw = density of particle and
water, respectively, kg/m3
Subscripts calc = calculated
exp = experimental max = maximal min = minimal p = for particle or in the presence of particles w = for water or in the presence of water Abbreviation TIS = two impinging streams
Literature Cited Elperin, I. T. “Heat and Mass Transfer in Impinging Streams”. Inzh. Fiz. Zh. 1961, 6, 62-68. Herskowitz, D.; Herskowitz, W.; Tamir, A. ”Desorption of Acetone in a Two-Impinging-Streams Spray Desorber.” Chem. Eng. Sci. 1987,42, 2331-2337. Hirose, T.; Mori, Y.; Sato, Y. “Liquid-to-Particle Mass Transfer in Fixed Bed Reactor with Cocurrent Gas-Liquid Downflow”. J . Chem. Eng. Jpn. 1976, 9, 220-225.
Kitron, A.; Buchman, R.; Luzzatto, K.; Tamir, A. “Drying and Mixing of Solids and Particles RTD in Four Impinging Streams and Multistage Two Impinging Streams Reactors”. Ind. Eng. Chem. Res. 1987, 26, 2454-2461. Luzzatto, K.; Tamir, A.; Elperin, I. “A New Two-Impinging-Streams Reactor”. AIChE J . 1984, 30, 600-608. Tamir, A. “Absorption of Acetone in a Two-Impinging-Streams Absorber”. Chem. Eng. Sci. 1986,41, 3023-3030. Tamir, A.; Grinholtz, M. “Performance of a Continuous Solid-Liquid Two-Impinging-Streams (TIS) Reactor: Dissolution of Solids, Hydrodynamics, Mean Residence Time, and Hold-up of the Particles”. Ind. Eng. Chem. Res. 1987, 26, 726-731. Tamir, A.; Herskowitz, D. “Absorption of COPin a New Two-Impinging-Streams Absorber”. Chem. Eng. Sci. 1985,40, 2149-2151. Tamir, A.; Kitron, Y. “Application of Impinging-Streamsin Chemical Engineering Processes”. Chem. Eng. Commun. 1987,50,241-330. Tamir, A.; Luzzatto, K. “Solid-Solid and Gas-Gas Mixing Properties of a New Two-Impinging-Streams Mixer”. AIChE J. 1985a, 31, 781-787. Tamir, A.; Luzzatto, K. “Mixing of Solids in Impinging-Streams Reactors”. J . Powder Bulk Solids Technol. 198Sb, 9, 15-24. Tamir, A.; Shalmon, B. “Scale-up of Two Impinging Streams (TIS) Reactors”. Ind. Eng. Chem. Res. 1988, 27, 238-242. Tamir, A.; Sobhi, S. “A New Two-Impinging-Streams Emulsifier”. AIChE J. 1985, 31, 2089-2092. Tamir, A.; Elperin, I.; Luzzatto, K. “Drying in a New Two-Impinging-Streams Reactor”. Chem. Eng. Sci. 1984, 39, 139-146. Tamir, A.; Luzzatto, K.; Sartana, D.; Salomon, S. “A Correlation Based on the Physical Properties of the Solid Particles for the Evaluation of the Pressure Drop in the Two-Impinging-Streams Gas-Solid Reactor”. AIChE J. 1985, 31, 1744-1746. Ziv, A.; Luzzatto, K.; Tamir, A. ”Application of Free ImpingingStreams to the Combustion of Gas and Pulverized Coal”. Combust. Sci. Technol. 1988, in press. Received for review November 5, 1987 Revised manuscript received April 21, 1988 Accepted May 3, 1988
Determination of the Local Voidage Distribution in Random Packed Beds of Complex Geometry Felix A. Schneider and David W. T. Rippin* Technisch-Chemisches Labor, ETH Zentrum, 8092 Zurich, Switzerland
The voidage fraction distribution in packed beds is visually displayed and recorded by a newly developed optical method that is fast and can be easily demonstrated. It can be used on fixed beds of more complex form than a single packed tube and aids in the understanding of the behavior of new types of reactor configuration. Results are reported for the voidage distribution within a single tube, which are in close agreement with these obtained by other, much more expensive and timeconsuming methods. New results are presented for voidage distribution in radial flow configurations not accessible by other methods. The implications for heat transfer in a radial flow reactor are discussed. 1. Introduction A number of different methods are described in the literature for the determination of the local voidage distribution in packed beds. They can be roughly divided into four classes (Figure 1). A. Incremental Filling. This procedure, first proposed by Shaffer (1952), was improved by Ridgway and Tarbuck (1966): during the stepwise filling of a rapidly rotating cylinder, the void fraction is deduced from the increase in level and volume. B. Solidification and Stepwise Removal. In this, the most commonly used method, the packing is immo* T o whom all correspondence should be addressed.
0888-5885/88/2627-1936$01.50/0
bilized with a slowly solidifying fluid. The resulting body is progressively reduced by turning in a lathe. To determine the voidage, either the waxy filling material is removed from the separated portions again or the proportions of the two materials of different density are deduced from the decrease of weight and volume (Roblee et al., 1958; Benenati and Brosilow, 1962; Goodling et al., 1983). C. Individual Measurement and Enumeration. Another group of experimenters determined the spatial location of the individual particles in the fixed bed and derived the local distribution by integration. Pillai (1977) used for this a two-dimensional bed of flat discs (Cl), and Schuster and Vortmeyer (1980) removed the particles of a packing one by one from above with the sticky end of a movable cylindrical sample probe (C2).
0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 10,1988 1937
C
C
2
V
Figure 2. Transparency of the packing studied. The figure shows a letter head below a 40-mm-thick layer of blue spheres of 5-mm diameter, the interstices being filled with liquid of the same refractive index.
Figure 1. Four methods of determining void fraction. (A) Shaffer (1952): incremental filling of a stationary cylinder. (B) Benenati and Brosilow (1962): filling and solidification in the void spaces; progressive removal and analysis of the solid body. (C.1) Pillai (1977): location of the centers ( 2 ) of two-dimensional discs. (C.2) Schuster and Vortmeyer (1980): removal of individual particles with a sticky probe and measurement of their position. (D) Thadani and Pebbles (1966): projection of X-rays through the packing ( x ) and scanning of the picture with visible light (u).
D. Projection of the Bed. There are various possibilities of passing light or other electromagnetic radiation through a fixed bed. These can be used to produce projections in which the varying transparency of the bed can be attributed to fluctuations in the voidage. In this way, photographs of the local average density can be produced. In 1966, Thadani and Pebbles used X-rays which could shine straight through sections of a packing. Despite the simplicity of the method, its use was somewhat laborious. Thus, the projection method proposed here, and described below based on visible light, was developed. Buchlin et al. (1977) used the same effect to obtain an optically homogeneous bed, but they then illuminated segments of the bed containing a fluorescent fluid with a laser in order to reveal the location of individual spheres in one cross section. The methods described to date all suffer from one or more of the following deficiencies: they require complex and costly equipment; the are laborious and time consuming; special materials must be used, which are destroyed during the measuring procedure by machining away; only systems with radial symmetry can be treated; no information is available on local or (for systems with radial symmetry) circumferential variability. The new method described here is clear, simple, and rapid; is nondestructive; provides results comparable with these obtained by earlier methods; and is not limited to systems with radial symmetry. 2. Basis of the Method A packing of transparent glass spheres in air does not itself appear to be transparent. The individual spheres act as optical lenses which refract the light in all directions. This effect arises from the different velocities of light in glass and air. It can be eliminated if the air is replaced by a fluid with the same optical properties as the solid. For normal glass, the refractive index used to characterize this
*V
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Figure 3. Construction of the experiment. In a glass vessel of 140-mm diameter, metal tubes (T) of from 16 to 55 mm diameter stand in a transparent packing of glass spheres (P) completely covered by the liquid (L). The experiment is illuminated from below (V)and photographed from above.
effect is approximately 1.52. Since it is dependent not only on the type of glass but also the wavelength of the light, fiitered, if possible monochromatic, lighting should be used. The same refractive index can be obtained experimentally by mixing toluene (n = 1.494) and benzyl alcohol (n = 1.538). The optical homogeneity which can be produced in this way is shown in Figure 2. The voidage distribution can be shown by coloring the liquid or using colored glass particles. When the bed is projected, the brightness of any point on the image is a measure of the light absorption by the colorant along the projection axis and therefore gives the average local voidage of the bed along this line. Even with an overhead projector, excellent definition can be achieved to demonstrate the effect. To obtain in this case good brightness and resolution, only slightly colored liquids and a modest bed thickness (30 mm) should be used. 3. The Experimental Procedure A high-walled glass vessel of approximately 140-mm diameter filled with liquid was set up on a Plexiglas plate and illuminated from below (Figure 3). In it were placed tube sections of height 50 mm and diameter 16-55 mm. Tube diameters are typical for tubes used to carry the
1938 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988
Figure 4. Projection of a tube with d t / d = 9.8. Tube of 50-mm internal diameter (dJ displayed as large bfack ring, blue spheres of 5.1 mm diameter (dp),50-mm height of projected packing. The light regions correspond to high local voidage.
catalyst in axial flow packed tubular reactors or for cooling tubes embedded in a packing. The height of the tube sections is sufficient to contain a representative depth of packing (- 10 layers). Increasing the height would increase the light scattering, which depends on the number of interfaces traversed by the transmitted light and could lead to loss of definition in the image. The refractive index was adjusted to that of the available glass beads by mixing toluene and benzyl alcohol. A first series of experiments was conducted with slightly blue-toned glass spheres of (5.05 f 0.07)-mm diameter in colorless liquid. A 1000-W spot light was used for indirect illumination with a red filter in front of the camera. For all other experiments, methyl red and crystal violet were selected to color the solution. These dyes, like many others, are not completely stable in light, but their decomposition products do not cloud the liquid. To reduce trapping of air bubbles, the glass spheres were tipped slowly into the model reactor previously filled with the liquid and the particles smoothed over. The fluid was sucked off to a level at which the whole model was still covered with an undisturbed surface. Stanek and Eckert (1979) found that the voidage structure of the bed was not significantly affected by whether the particles were filled wet or dry. On illumination, the density patterns were clearly visible and could have been recorded immediately with a photo cell. For the required results, however, it was sufficient to make photographs with a 200-mm teleobjective and to accept the projection errors arising way from the center.
Figure 5. Projection of a tube with d+,/dp= 4.1. Tube of 21-mm internal diameter, blue spheres of 5.1-mm diameter, packing height 50 mm.
Figure 6. Radial distribution of relative intensity of photograph Figure 4 (dt/dp= 9.8) with strongly damped circumferential fluctuations. The outermost voidage maxima (HI)appear too weak since they are overlaid by the neighboring tube walls ( t )due to projection errors and eccentricity.
4. Verification
Measurements were made first with spherical particles inside cylindrical tubes to permit comparison with results of other workers and confirm the validity of the method. For the tube sizes used with the 5-mm spheres, the tubeto-particle diameter ratio covered a range from 3 to 10, corresponding to values found in industrial practice. The photographs of different tubes filled with blue particles (e.g., Figures 4 and 5) show regular patterns, with the dark regions correspondingto lower voidage. A more quantitative representation of the voidage distribution as a function of radial position could be obtained by integrating around a circle concentric with the tube for each radius. The same effect was achieved by mounting the photograph in a simple apparatus which rotated the photograph about the point corresponding to the center of the tube. A photocell was mounted so that its radial position from the point corresponding to the center of the
Figure 7. Radial distribution of relative intensity of photograph Figure 5 (d,/d, = 4.1) with weakly damped circumferential fluctuations. With increasing distance from the tube wall ( t ) ,the variability increases until it declines again on approaching the center ( 2 ) due to decreasing area of the sample.
tube could be varied. Curves representing the voidage distribution were obtained by scanning the rotating photograph across the diameter of the tube with the photocell, (Figures 6 and 7). The curves can be arranged to show, depending on the inertia of the recorder and the speed of
Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 1939
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Figure 8. Comparison with literature data. The radial distances from tube wall (h) of minima (marked points) and maxima (small points) of the oscillating voidage distribution in the cited publications ((A) Benenati and Brosilow, 1962; (B) Goodling et al., 1983) were measured and compared with current results (C) for different tube diameters (dJ. All scales are relative to particle diameter (dJ, and the dotted lines show the calculated minima of the most dense packing.
rotation, either the circumferential fluctuations of voidage superimposed on the general trend (Figure 7) or only the average values (Figure 6). The locations of maxima and minima in the voidage distributions identified from these curves are compared in Figure 8 with the values read from published curves. The results agree very well with those of other authors. The various sources of error (influence of the packing by the base plate, unevenness of the packing height, impurities and air bubbles in spheres or liquid, and also eccentricity and parallax errors) are sufficiently small and do not affect the results significantly. To obtain quantitative results, it would be desirable to make the absorption measurements directly on the experimental equipment and to take a calibration curve. Parallax errors could be avoided completely by scanning the system with a photocell positioned directly below a monochromator. 5. Second Series of Experiments: Voidage Distribution around Tubes Immersed in a Packed Bed The new method was used to examine the voidage distribution in the space surrounding a regular array of tubes-a configuration not previously studied. 5.1. Background. Heterogeneous gas-phase catalysis is of great economic importance. Many process reactions are strongly exo- or endothermic,which makes it necessary to describe the heat flow in the reactor as accurately as possible. Even the normal assumptions of plug flow for fixed beds, as considered for example in the textbook of Bird et al. (1960), can lead to very demanding models (Wellauer, 1985). The long-recognized irregularities in the flow near the boundaries of fixed beds were newly investigated by Schuster and Vortmeyer in 1981. In 1982, Vortmeyer and Winter extended their model for the description of heat transfer and thus obtained a further improvement in the simulation of fixed bed reactions. The flow irregularities were attributed to the irregular distribution of the voidage at the boundary of a fixed bed. 5.2. Objective. In a larger project, the heat-transfer properties of a nonadiabatic radial flow reactor (RFR) (Ohsaki et al., 1980) are being compared with those of a conventional multitubular axial flow reactor (AFR). It is of interest to identify differences in the voidage distribution.
Figure 9. Voidage distribution in RFR I. Between the five tubes of 50-mm length and of 27-mm external diameter, typical triangular bed spaces of the RFR are built up, transverse to the flow direction. The packing of clear 6-mm spheres in a colored solution shows dark zones corresponding to high local voidage.
Figure 10. Voidage distribution in RFR 11. Tubes 50 mm long and 27-mm external diameter transverse to the flow direction together with a packing of clear 3-mm spheres in a colored solution.
5.3. Experiment. The bed of the radial flow reactor is like a tubular heat exchanger in which the spaces between the tubes are filled with catalyst packing. Thus, in contrast to the tubes of the AFR, the RFR has no rotational symmetry. The best known methods for determination of the voidage distribution in tubes and at walls can therefore not be used. Since the method of Thadani and Pebbles (1966) using X-rays seemed too laborious, the new method was used. Figures 9 and 10 show photographs of sections of the reactor taken in the plane of fluid flow. 5.4. Results. 5.4.1. Structure. From examination of the photographs, general characteristics of the voidage distribution in spaces between the tubes of the RFR can be identified. The packing density in the RFR oscillates in the neighborhood of the wall just as for single surfaces. Around each tube, a regular hexagonal zone can be found, where the packing develops undisturbed rings. In the corners of these zones, stacks of single spheres and other simple packing geometries can be observed just as in the center of single tubes (Figure 11). 5.4.2. Velocity Distribution. A local gas velocity distribution in the bed could be obtained from the above model by using a finite element method, recollecting that in the RFR the flow distribution is across the bank of tubes. Generally, according to Schuster and Vortmeyer (1981), it can be said that in a narrow zone along the walls the gas can flow considerably faster. For the AFR, the width of this zone is of the order of half a particle diameter (Martin, 1978).
1940 Ind. Eng. Chem. Res., Vol. 27, No. 10,1988 IT
,FR
RF
RFR
Figure 11. Model of the packing structure. Each of the hexagonal cells (h) making up the RFR tube bundle corresponds, in respect to its voidage distribution, to the same section around an individual tube (IT).
5.4.3. Comparison of Heat-Transfer Properties. In the AFR, gas can flow through the whole reactor along the wall of the same tube in a region with large voidage. The high velocity of this bypass and the small contact area it has with the solid particles impair the heat-transfer properties of the whole bed (Martin, 1978). This effect, together with local irregularities in the interior of the bed, accounts for the fact that the Nusselt number is only about two-thirds of the value for flow around a single sphere outside the bed (Gnielinski, 1978). However, in the RFR, this effect will be reduced; the bypass zones are not continuously parallel to the direction of flow, so after each row of tubes, the fluid stream must flow through the more dense interior of the bed (Figure 12). The different streams which are cooled to different extents are mixed more intensively, so the temperature gradients which arise in the bed should be smaller. This difference would indicate better heat transfer at the same pressure drop. However, other relevant differences between AFR and RFR, such as the possible formation of stagnant regions, must certainly also be considered before better properties can definitely be postulated. 6. Conclusions A simple method has been presented to display the local voidage in a packed bed, which can then be scanned in any desired way. Observations in a single tube closely reproduce the oscillatory voidage distributions obtained by other methods. Some advantages and interesting features of the method are as follows. Wide Range of Possible Configurations. The method is not limited to systems with rotational symmetry. So observations of the voidage structure within a bed surrounding an array of tubes suggest that this configuration may have favorable heat-transfer characteristics. The optimal conditions for the experiment are the following: particle diameter is 3 mm or more; particle shape must not contain concave regions to avoid trapping of air bubbles; projection depth should be 100 mm maximum and about 30 mm for overhead projector; the maximum resolution that can be achieved by the projection depends on the optical homogeneity of the transparent packing and the spectral range of the light source. Speed. A single experiment can be assembled and photographed in about 20 min. Thus, large experimental designs can be carried out such as are necessary to examine mixed packings.
*
*
Figure 12. Location of bypasses in AFR and RFR. The light gray zones adjacent to the tube walls (t) act as bypass channels (arrows) due to lower particle density than the central core (dark gray).
Illustrative Value. For demonstrations, the experiment can be carried out very simply using an overhead projector. The results are optically attractive and readily explained. They can contribute to the better understanding of the many sided problems of describing fixed beds. Literature Cited Benenati, R. F.; Brosilow, C. B. “Void Fraction Distribution in Beds of Spheres”. AIChE J. 1962,8,359-361. Bird, B. R.; Stewart, W. E.; Lightfoot,E. N. Transport Phenomena; Wiley: New York, 1960;p 279. Buchlin, J. M.; Riethmuller, M.; Ginoux, J. J. “A Fluorescence Method for the Measurement of the Local Voidage in Random Packed Beds”. Chem. Eng. Sei. 1977,32,1116-1119. Gnielinski, V. “Gleichungen zur Berechnung des W h m e und Stoffuberganges in durchstriimten ruhenden Kugelschiittungen bei mittleren und grossen Peclet-zahlen”. Verfuhrenstechnik 1978,12,363-366. Goodling, J. S.; Vachon, R. I.; Stelpflug, W. S.; Ying, S. J. “Radial Porosity Distribution in Cylindrical Beds Packed with Spheres”. Powder Technol. 1983,35,23-29. Martin, H. “Neue Erkenntnisse uber den Wiirme und Stoffiibergang in durchstromten Haufwerken bei niedrigen Peclet-Zahlen”. Chem.-Ing.-Tech. 1978,50,116-118. Ohsaki, K.; Kobayashi, J. Z. Y.; Watanabe, H. “Process and Apparatus for Reacting Gaseous Raw Material in Contact with a Solid Catalyst Layer”. UK Patent Application GB 2 046618A,1980. Pillai, K. K. “Voidage Variation at the Wall of a Packed Bed of Spheres”. Chem. Eng. Sei. 1977,32,59-61. Ridgway, K.;Tarbuck, K. J. “RadialVoidage Variation in Randomly Packed Beds of Spheres of Different Sizes”. J.Pharm. PhurmaCOZ.1966,18,168S-l75S. Roblee, L. H.; Baird, R. M.; Tierney, J. W. “Radial Porosity Variations in Packed Beds”. AIChE J. 1958,4,460-464. Shaffer, M. R. M.S. Thesis, Purdue University, West Lafayette, IN, 1952. Schuster, J.; Vortmeyer, D. “Simple Procedure for approximate Determination of Porosity in Packings as a Function of Distance from Wall” (German). Chem.-Ing.-Tech. 1980,52,848-849. Schuster, J.; Vortmeyer, D. “Geschwindigkeitsverteilung in gasdurchstriimten, isothermen Kugelschuttungen”. Chem.-Ing.Tech. 1981,53,806-807. Stanek, V.; Eckert, V. “Study of the Area Porosity Profiles in a Bed of Equal-Diameter Spheres Confined by a Plane”. Chem. Eng. Sci. 1979,34,933-940. Thadani, M. C.; Peebles, F. N. “Variation of Local Void Fraction in Randomly Packed Beds of Equal Spheres”. Ind. Eng. Process Des. Dev. 1966,5,265-268.
I n d . Eng. Chem. Res. 1988,27, 1941-1946 Vortmeyer, D.; Winter, R. P. “Impact of Porosity and Velocity Distribution on the Theoretical Prediction of Fixed-Bed Chemical Reactor Performance-Comparison with Experimental Data”. ACS Symp. Ser. 1982,196, 49-61. Wellauer, T. P., ’Optimal Policies in Maleic Anhydride Production
1941
through Detailed Reactor Modelling”. Ph.D. Dissertation 7785, ETH Zurich, Switzerland, 1985.
Received for review November 5, 1987 Accepted April 28, 1988
Structure and Porosity of Simulated Sediments of Polydisperse Particles Eric Dickinson* Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, U.K.
Steven J . Milne and Mohamed Pate1 Department of Ceramics, University of Leeds, Leeds LS2 9JT, U.K.
Computer simulation is used to study the effect of particle-size distribution on the microstructure of packed beds of hard non-Brownian particles in two dimensions. For monodisperse, polydisperse, and bimodal distributions, we report data on packing densities, pore-size distributions, and extents of particle ordering. It is found that the presence of a relatively small degree of polydispersity gives simulated sediment structures which are much more uniform in packing density than those formed from monodisperse suspensions. In relation to the formation of dense compacts for ceramic fabrication, it appears that a compromise has to be reached between the small pores of nearly monodisperse systems and the dislocation-free uniform structures of moderately polydisperse systems. The packing of spherical particles into a consolidated body is a process that is common to many fields of science and engineering. Amongst the many factors affecting the final density and porosity of the consolidated body, the distribution of sizes of the constituent particles is one of the most important. Experimental studies of particle packing go back many years-see, for example, the work of Westman and Hugill (1930), McGeary (1961), and Owe Berg et al. (1969). More recently, however, developments in computer simulation methods have proved increasingly valuable in helping to elucidate the principles which underlie packing and consolidation processes (Finney, 1970; Berryman, 1983;Jodrey and Tory, 1985; Gotoh et al., 1986; Ouchlyama and Tanaka, 1986; Rosato et al., 1987). Current efforts to improve the techniques of ceramic fabrication (Brinker et al., 1984) have made ceramicists aware of a need to gain better control over the critical powder compaction procedures. Fabrication generally involves first compacting a powder and then sintering it to convert the consolidated body into a dense ceramic material (Norton, 1974; Kingery, 1978). The properties of the resulting ceramic are related to its microstructure (i.e., grain-size and pore-size distribution), which in turn depends on the characteristics of the starting powder and the way the particles are arranged in the compact. The stringent demands being placed on modern engineering ceramics imply therefore a high degree of control over the final microstructure. Against this technological background, the fundamental chemical and physical aspects of fabrication procedures are currently being reexamined in order to attain more uniform powder compacts and hence more consistency and reliability in ceramic microstructures and properties. One way to pack particles is to allow them to settle out under gravity (or in a centrifuge). Computer simulation of sedimentation processes enables the various factors affecting the structure and properties of sediments to be investigated independently and systematically. In a recent Brownian dynamics simulation of sediment formation by irreversible single-particle deposition (Ansell and Dickinson, 1986), it has been shown how the sediment structure depends on the sedimenting field strength and on the *To whom correspondence should be addressed. 0888-5885/88/2627-1941$01.50/0
nature of the hydrodynamic and colloidal interactions between the settling particle and the existing sediment. Aggregates and sediments formed from “sticky” (coagulating) particles are of low particle density and high porosity, since there is a strong tendency purely on statistical grounds for the newly accumulating particles to stick to protruding arms of the growing aggregate or sediment, in preference to filling up gaps in the interior of the aggregate or sediment structure (Ansell and Dickinson, 1986,1987). In order to produce a compact sediment, it is necessary to have “nonsticky“ particles which do not stick irreversibly at the first point of contact with the existing sediment but instead fill up holes in its developing structure. This paper describes a simulation study of sediment formation from a dilute polydisperse suspension of nonsticky hard particles. In order to focus attention totally on the effects of particle-size polydispersity, we consider here a simple two-dimensional model in which Brownian motion and hydrodynamic interactions are excluded from the calculations. Our aim is to determine how sediment structure and porosity are related to the degree of polydispersity and to assess the implications of the results for the optimum choice of particle-size distribution for ceramic processing. One of the authors has recently described (Dickinson, 1986) how the equilibrium structure of a colloidal suspension of spherical particles is affected by polydispersity. In particular, it has been shown by computer simulation (Dickinson et al., 1981) that the crystalline ordered arrangement of a monodisperse system is disrupted, at constant particle volume fraction, by the introduction of polydispersity. Above a certain critical degree of polydispersity, estimated at about 30% (Dickinson and Parker, 1985; Pusey, 19871, it is not possible to form an ordered equilibrium structure at all. Similar considerations are expected to apply to the polydisperse sediment structures simulated in this present study, even though the resulting structures are not, of course, at equilibrium in the thermodynamic sense. We are interested therefore in the effect of polydispersity on the extent of long-range ordering, as well as on its effect on the pore-size distribution. Simulation Model and Methodology Packed beds are simulated in two dimensions by depositing circular particles one at a time in a cell with a rigid 0 1988 American Chemical Society
