RETRACTED ARTICLE: Constriction Size Reduction Approach in Granular Filters

Abstract

In order to assess the capability of a filter skeleton to retain flowing particles transported from eroded soil, different models based on continuous or discrete medium are available. The porous medium is often described by the grain size distribution, whereas constriction size distribution (CSD) is the key parameter governing the filtration process. This study describes the filter constrictions analysis and its application to the reduction after filtration. This investigation involves combined hole erosion-filtration experiments describing internal erosion of a base soil and particles filtration by a granular medium. The combination of experimental data of measured porosity and analytical results of the CSD was used to evaluate constrictions size reduction subsequently to filtration mechanism. The filtration depth was estimated according to retained particle mass and porosity reduction, which is evaluated from the evolution of hydraulic conductivity. The analysis of obtained results showed the occurrence of a non-uniform constriction reduction, suggesting an effective filtration depth.

Appendices

Appendix 1: Particle frequency by surface area (P _SA) of the filter particles in classes i;

If a filter material is composed of n diameters: D₁, D₂, D₃ …, D_n and their mass frequencies are P_m1, P_m2, P_m3, …, P_mn, respectively, then their respective frequencies by surface area P_SAi can be obtained as follows (Humes 1996 cited in Dallo and Wang 2012).

Consider that the specific gravity of the filter particles are the same and:

• Volume of filter particles = V.

• Mass/volume frequency of filter particles in the class i (as shown in Fig. 

  Fig. 15

  

  Discretization of the filter grains size curve

  15) = P_mi.

• Total volume of filter particles in the class i  =  $P_{{{\text{mi}}}} \times V$.

• Total number of filter particles in the class i, $N_{i} = \frac{{P_{{{\text{mi}}}} \times V}}{{\pi D_{i}^{3} /6}}$

• Total surface area of the particles in the class i:

  $$S_{{{\text{Ai}}}} = N_{i} \left( {\pi D_{i} } \right) = \frac{{P_{{{\text{mi}}}} \times V}}{{\left( {\frac{{\pi D_{i}^{3} }}{6}} \right)}}\left( {\pi D_{i}^{2} } \right) = 6 \times V\frac{{P_{{{\text{mi}}}} }}{{D_{i} }}$$

• Total surface area of the filter particles in total volume $V: = \mathop \sum \limits_{i = 1}^{i = n} ({\text{SA}}_{i} ) = 6V\mathop \sum \limits_{i = 1}^{i = n} \left( {\frac{{P_{{{\text{mi}}}} }}{{D_{i} }}} \right)$

• Particle frequency by surface area of the filter particles in classes i:

  $$P_{{{\text{SAi}}}} = \frac{{{\text{SA}}_{i} }}{{\mathop \sum \nolimits_{i = 1}^{i = n} {\text{SA}}_{i} }} = \frac{{\frac{{P_{{{\text{mi}}}} }}{{D_{i} }}}}{{\mathop \sum \nolimits_{i = 1}^{i = n} \frac{{P_{{{\text{mi}}}} }}{{D_{i} }}}}$$

The particle size distribution based on surface area can be determined by equation:

$$P_{{{\text{SAi}}}} = \frac{{\frac{{P_{{{\text{mi}}}} }}{{D_{i} }}}}{{\mathop \sum \nolimits_{i = 1}^{i = n} \frac{{P_{{{\text{mi}}}} }}{{D_{i} }}}}$$

Appendix 2: Determination of angles $\alpha ,\beta ,\gamma et \delta$ for the calculation of S _v areas

In this case, the angle α is considered that corresponding to the largest particles in the group, the angles β, γ et δ are to be calculated (Fig. 

Fig. 16

The two extreme cases of the angle α

16). The area of constrictions S_V is calculated by:

$$S_{{\text{V}}} = \frac{{a*d*{\text{sin}}\left( \alpha \right)}}{2} + \frac{{b*c*{\text{sin}}\left( \gamma \right)}}{2} - \frac{1}{8}\left( {D_{i}^{2} \alpha + D_{j}^{2} \beta + D_{k}^{2} \gamma + D_{{\text{m}}}^{2} \delta } \right)$$

The given angle α suffices to fix the geometric arrangement of the system. The angles β, γ and δ can therefore be expressed as functions of the single parameter α. After explaining these dependencies, it suffices to vary the angle α between 0 and π by an increment of (2°), the iteration continues, and to recover the maximum value of S_v obtained over this interval.

The area of the triangle (D_j D_k D_m) is equal to:

$$A_{{D_{j} D_{k} D_{{\text{m}}} }} = \frac{{b*c* {\text{sin}}\left( \gamma \right)}}{2}$$

Expression of the angle δ as a function of α:

$$\delta = 2\pi - \alpha - \beta - \gamma$$