Tortuosity variation in a low density binary particulate bed

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Abstract

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 ɛ in the form T = 1/ɛn, 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 xD 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.

Introduction

Mixed beds of particles have a wide application in industry and sciences. Granular beds and, in particular, packing of particles with different sizes display a wide range of values of the porosity (ɛ) and pore tortuosity (T) [1], [2], [3], [4], [5]. Models of the binary particle beds porosity [6], [7], [8], [9], [10], [11], [12], [13], [14] and permeability [15], [16], [17], [18], [19] versus the volume fraction of the mixture components have been thoroughly described.

The tortuosity is defined as T = Le/L, where Le is the average flow pathway length and L is the bed thickness. Investigations on T are concentrated on the establishment of a relationship between the overall porosity and tortuosity [20]. For granular packings, the main effort has been focused on the determination of a fixed tortuosity value [14], [21], [22].

For a mixed bed of particles significantly different in size, to take into account the influence of the porosity on the permeability through the tortuosity is of major importance [16]. Due to the different methods applied for packing preparation, the values of bed porosity lie between “loose” and “dense” packing values, making of key importance to know how tortuosity is related with packing porosity.

Among the proposed relations describing the relationship T versus ɛ [23], [24], [25], [26], [27], for granular packings, a power law relationship is the most frequently used:T=1εnwhere n is a numerical value.

There are many evidences that n depends on the packing properties. For binary mixtures, Klusácek and Schneider [28] admitted that n is not a constant. Assuming that in a porous medium there are m classes of pores and that each class occupies the same portion of the total porosity, Millington and Quirk [29] suggested, for unconsolidated systems, n to be 1/3. Zhang and Bishop [30] and Mota et al. [31] applied n = 0.5. For loose packed spherical particles mixtures the best fit of T versus ɛ was obtained for n = 0.4 [27], [32]. n = 0.4 gives a good approach for tortuosity (1.47 average value) measured in a spheres packing with porosity between 0.363 ± 0.030 [21]. Based on these observations, it may be speculated that the exponent index for granular beds describing the dependence of tortuosity on porosity ranges from 0.4 (loose packing) to 0.5 (dense packing).

The above-mentioned assumption can be confirmed by the data of Currie [1] (for sphere mixtures; sand mixtures; spheres/sand mixture) plotted in Fig. 1 together with the plot of Eq. (1) at n = 0.4 and 0.5. As can be seen, most of the data lies between the two functions.

Mota et al. [33] investigated a binary mixture spheres at different olume fractions of large particles xD for particle ratios D/d of 13.3, 20, and 26.7. Experiments with the binary particulate bed show that the dependence of n on the fractional content xD lies in the range of 0.5 (for the monosize packing) up to ∼0.4 (in the region of the minimum porosity of the binary bed).

A justification for the variation of the parameter n in Eq. (1) is required with the purpose of establishing a relationship between the binary packed bed porosity, fractional content, tortuosity and, hence, the permeability.

Section snippets

Experimental basis

Loose packed binary mixtures of glass beads of a size ratio from D/d = 1 up to 53.8 were analysed in terms of porosity, tortuosity and permeability. Most of the experimental data has been obtained in previous works [27], [33] and additional experiments for high D/d ratio were performed using previously described procedures [33], [34], [35]. The packing procedure was performed without intensive compacting methods e.g. vibration, etc.

Packing porosity was measured by the volumetric method, whereas

Analysis and discussion

Data presented in a previous work [33] shows that the parameter n is a function of xD at the size ratio D/d = 13.3, 20, and 26.7. Measurements were made for mixtures with xD from 0 up to xDmin, this value corresponding to a minimum packing porosity ɛmin (xDmin was around 0.65–0.7). It was shown that the particle arrangement in the binary mixtures at ɛmin is characterised as a loose packing density. As a result, the tortuosity becomes lower being this reflected in the reduction of the parameter n

Conclusion

The complexity of the processes involved in the formation of granular beds results in the inter-dependence of the main parameters included in the permeability: packing porosity and tortuosity. The bed porosity in the region of ɛmin is affected by particle size ratio and packing fractional content.

The obtained results show that the parameter n in the tortuosity formula T = 1/ɛn is a function of the packing content xD and lies in the range 0.4–0.5. The reason for n variation may be explained by the

Acknowledgements

The authors wish to thank Fundação para a Ciência e Tecnologia (FCT) for having provided the funds to perform this work through the project POCTI/EQU/37500/2001, as well as for the grant accorded to A. Yelshin. This project was partially funded by FEDER.

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