Tortuosity variation in a low density binary particulate bed
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:where 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.
References (36)
- et al.
Powder Technol.
(1971) Chem. Eng. J.
(1975)- et al.
J. Colloid Interface Sci.
(1999) - et al.
Chem. Eng. Sci.
(2003) J. Colloid Interface Sci.
(1999)- et al.
Chem. Eng. Sci.
(1981) - et al.
Colloids Surf.
(1987) - et al.
Chem. Eng. Sci.
(1997) - et al.
Chem. Eng. Sci.
(1981) - et al.
Water Res.
(1994)
Sep. Purif. Technol.
Br. J. Appl. Phys.
Mass Transfer in Heterogeneous Catalysis
Dynamics of Fluids in Porous Media
Fundamentals of transport phenomena in porous media
Ind. Eng. Chem. Fundam.
Ind. Eng. Chem. Res.
Cited by (82)
Modelling swelling effects in real espresso extraction using a 1-dimensional coarse-grained model
2024, Journal of Food EngineeringOn the hydrodynamics of macroporous structures: Experimental, CFD and artificial neural network analysis
2023, Chemical Engineering Journal AdvancesA mesoscale model for heterogeneous reactions in porous media applied to reducing porous iron oxide
2023, Chemical Engineering ScienceInfluence of particle size distribution on espresso extraction via packed bed compression
2023, Journal of Food EngineeringCitation Excerpt :An increment in the tortuosity of the bed also leads to a rise in permeability. Dias et al. (2006) proposed an expression of this parameter as an inversely proportional function of the bed porosity. The latter could justify the obtained exponent for the dry bed porosity in Equation (11).
A numerical investigation on the hydrogen reduction of wüstite using a 2D mesoscale method
2022, International Journal of Hydrogen EnergyNumerical study of the hydraulic tortuosity for fluid flow through elliptical particle packings
2022, Powder TechnologyCitation Excerpt :The LBM-DEM model is then validated by comparing the simulated Th for mono-size elliptical particle packing systems and experimental permeability (K) for porous media with different pore sizes. A generalized Th format containing the ε and r is proposed based on the format from the Bruggeman correlation [25]. Then LBM simulations are conducted for formulating the Th correlation for elliptical particle packings with different r. Finally, the newly proposed Th correlation is verified with previous literature results.