A mathematical model of aluminum depth filtration with ceramic foam filters: Part II. Application to long-term filtratio
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I. INTRODUCTION
THE importance of understanding the phenomena occurring during long-term filtration is basically to predict the filter performance and overall filtration efficiency under specific conditions. During this process, the filter accumulates inclusions decreasing its effective porosity, which may lead to clogging or filter aging. The models to determine the initial filtration efficiency represent a first approximation to obtain the long-term filtration efficiency but do not offer any rationale of the aging process. The depth filtration theory,[1] established from results of filtration in granular media, indicates that particle accumulation forms a smooth coat on the filter grains improving the filtration efficiency by reducing the distance from the traveling particles to the collector wall. However, at a higher level of particle accumulation, the clogging of necks or windows between pores becomes more important to determine the filtration efficiency, reducing its value by incrementing the interstitial fluid velocity. The transition from smooth coat to the blocking stage has been observed to occur when the specific deposit, s, defined as the volume of the particles deposited per unit volume of filter, has a value from 0.04 to 0.06. Therefore, the filtration coefficient l reaches a maximum at s 5 0.05, approximately. In order to quantify this observation, Tien[1] has combined F.A. ACOSTA G., Assistant Professor, and A.H. CASTILLEJOS E., Professor, are with the Centro de Investigacio´n y de Estudios Avanzados del IPN-Unidad Saltillo, Apdo. Postal 663, Saltillo, Caoh., 25000 Me´xico. Manuscript submitted September 3, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS B
two categories of depth filtration models: an external model to compute l during the smooth coat stage and an internal model to evaluate the filtration coefficient along the clogging stage. The external model, in which the unit cell is a sphere acting as a collector grain, predicts an increment in l during filtration as a consequence of incrementing its grain radius by the smooth coat of trapped particles; the internal model, having a constricted tube as a unit cell, predicts that l declines with further filtration as a result of having blocked necks, forcing a high interstitial fluid velocity through the remaining necks. This approach introduces an empirical constant and particle deposit parameters in order to obtain results in agreement with measurements. Therefore, the computed results agree generally well with measurements except at the transition between stages, as previously mentioned, where l has an error of nearly 100 pct with respect to the observed value, and the computed pressure drop differs by up to a factor of 3 with respect to measured values. Wang et al.[2] used a spherical collector to study long-term filtration and introduced two concepts, namely, the shadow effect created by the deposited particles and the singular and random nature of the approaching particles. They calculated the efficiency of a spherical collector when particl
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