Binary mixtures with significant size ratios are scarcely studied. Yet, contaminants of chromatographic columns or ion-exchange resins have size ratios of δ < 0.1. Binary mixtures of glass beads with δ ) 0.1-0.0375 were used experimentally to measure packing porosity. Simultaneously, a significant number of published data was analyzed. A linear mixing model was adopted to predict the porosity of each particle fraction in the binary mixture. Deviations from the model may be caused by wedging of small particles between the large ones. Large particles may disturb the porous medium properties by inducing a wall effect over the small particles. Wedging analysis led to the conclusion that, for δ < 0.01, its effect is insignificant. The wall effect yields an additional void around the large particles as long as δ > 0.0035. For δ < 0.0035, the small particles form a monosized dense packing and both wedging and wall effects become negligible.

The importance of particle size ratio and particle composition in the properties of a mixed bed is well known. Nevertheless, the dependence of the bed channel tortuosity T on the porosity ε in the form T = 1/ε n , where n is assumed to be a constant, shows that the value of n depends on the properties of the packed bed. For loose packing, experimental data for binary mixtures of glass beads of a size ratio from 1 up to 53.8 was analysed in terms of porosity, tortuosity and permeability. The packing procedure was performed without intensive compacting methods e.g. vibration, etc. Obtained results show that the parameter n is a function of the volume fraction of large particles x D and, for spherical particles, lies in the range 0.4-0.5. The explanation for this variation is (1) a distortion effect on the small particles arrangement occurring near the large particle surface; (2) in the region of minimum porosity, near contact points of large particles, the occurrence of dead zones that are free of small particles. A relationship accounting for this effect is proposed that may be useful for the analysis of transport phenomena in granular bed filters, chromatographic columns, etc.

The problems related to tortuosity variation whenever filter cakes are composed of cells with different shapes and compressible biosuspensions are discussed. Presented examples show that neglecting the tortuosity variation may lead to significant miscalculations of cake porosity or of specific cake resistance. Specific cake resistance of rod-like particles in cross-flow filtration depends on the higher tortuosity obtained by the shear-induced ordered arrangement. In turn, spheroid cells such as baker's yeast cells do not affect tortuosity as much as the rod-shaped cells. By including tortuosity as a parameter of compressible cakes, a more precise representation of cakes' behaviour may be obtained. The tortuosity becomes a highly significant parameter with the increase in filtration pressure.

Diffusion in pure gels and gels with immobilized cells was analyzed. A model of diffusion assuming a homogeneous cell distribution in gel was improved by introducing a tortuosity value. By theoretical analysis and numerical modeling it was shown that the tortuosity of a gel with immobilized cells is the product of two factors: (1) tortuosity generated by the cells, T c , and (2) tortuosity of the gel matrix, T g , both variables being a function of cell volume fraction, φ c . Total tortuosity is thus T Σ ) T c T g . On the basis of this approach, it was possible to analyze diffusivity data for gels with immobilized cells. It was shown that, in these systems, the diffusivity η ) D e /D 0 is a complex function of (1) diffusivity in the gel, η g , and (2) diffusivity in immobilized cells, η c . The developed model allowed for the description of the dependence of D e /D 0 on φ c . Comparison with numerous published experimental data showed a good fit. Observed deviations might be explained by nonhomogeneous cell distributions inside the gel matrix.

In solid-liquid separation the knowledge of solids packing structure is important to control permeability and dewaterability. In particular, cakes formed in filtration are often represented by the composition in coarse and fine particles. In this work cakes were modelled by mixing a bed of coarse (spheres) and fine (kieselguhr of three types and kieselgel) particles with a wide size distribution, in order to obtain beds with different proportions of plate and rod-like particles. Size ratio of glass beads to kieselguhr particles were in the range 23-30. Porosity and permeability were measured for a range of large particle fraction in the mixture from 0 up to 1.0. The fractional porosity of each particle fraction was introduced as a parameter. The approach proposed in this work was also successfully applied to different published filtration data. It was found that (1) the presence of more than 10% of fines in the coarse granular bed significantly reduces the cake permeability; (2) to improve cake permeability the volume fraction of filter aid in suspension must be at least 50-60% of total solid volume; (3) obtained data may be used to control the porosity of a mixture, if the fractional porosity of large and small particles is known or to estimate mixture tortuosity.

Two-dimensional simulations of packed beds composed of binary and ternary particle mixtures were made and image analysis of the bed structure was used to determine the bed porosity and tortuosity. Both the porosity and tortuosity were found to be dependent upon the volume fraction of the large particles. However, the volume fraction alone does not totally determine either the porosity or the tortuosity. For a bed of two different sizes of particles, its tortuosity may be considered a product of two quantities, the macro-and micro-tortuosity, each of which can be determined from the corresponding monosized particle beds. 0 1999 Published by Elsevier Science B.V. All rights reserved.

In the region of minimum porosity of particulate binary mixtures, heat exchange and permeability were found to be higher than the ones obtained with a mono-size packing built with the same small size particles used in the binary packing. This effect was noticed in the range of the particles size ratio 0.1-1.0.The obtained improvement on thermal performance is related to the increase of effective thermal conductivity (ETC) in the binary packing and to the increase in transversal thermal dispersion due to the porosity decrease and tortuosity increase.Permeability can increase by a factor of two, if the size ratio between small and large spheres of a loose packing stays in the range 0.3-0.5.

SummaryThe permeability of binary packings of glass beads with different size ratio -13.3, 20, and 26.7, was investigated. In the Kozeny-Carman equation, the dependence of the tortuosity s on the mixture porosity e(x D ) was described according to s = 1/e n for different volume fraction of large particles in the mixture, x D . Obtained data on packing permeability shows that the parameter n is a function of the volume fraction and particle size ratio, with values between 0.5 and 0.4. This can be explained by the wall effect resulting from the arrangement of the small particles occurring near the large particle surface. A model taking in account this effect was suggested that can be useful in the characterization of transport phenomena in granular beds as well as in engineering applications. ª