Deep bed filtration : mathematical models and observations

Numerous mathematical models have been developed to evaluate both initial and transient stage removal efficiency of deep bed filters. Microscopic models either using trajectory analysis or convective diffusion equations were used to compute the initial removal efficiency. These models predicted the removal efficiency under favorable filtration conditions quantitatively, but failed to predict the removal efficiency under unfavorable conditions. They underestimated the removal efficiency under unfavorable conditions. Thus, semi-empirical formulations were developed to compute initial removal efficiencies under unfavorable conditions. Also, correction for the adhesion of particles onto filter grains improved the results obtained for removal efficiency from the trajectory analysis. Macroscopic models were used to predict the transient stage removal efficiency of deep bed filters. The O’Melia and Ali1 model assumed that the particle removal is due to filter grains as well as the particles that are already deposited onto the filter grain. Thus, semi-empirical models were used to predict the ripening of filtration. Several modifications were made to the model developed by O’Melia and Ali to predict the deterioration of particle removal during the transient stages of filtration. Models considering the removal of particles under favorable conditions and the accumulation of charges on the filter grains during the transient stages were also developed. This article evaluates those models and their applicability under different operating conditions of filtration.

Surface coverage of filter medium in deep bed filtration : mathematical modelling and experiments

This paper highlights the importance of surface coverage in modeling the removal of particles in deep bed filtration. A model that considers the saturation of sites on which particle deposition occurs is used. Experimental results obtained with monodispersed suspensions of 0.46 and 0.816 μm latex particles at different influent concentrations and ionic strengths were used to calculate the fraction of filter grain surface (β1) on which actual particle deposition occurs. This will be useful in evaluating the filter performance in terms of the utilization of available surface area of the filter medium. Further, the level of dendrite formation of particles on filter grains during filtration is expressed in terms of β1 and the specific surface coverage, θT (the fraction of a filter grain surface that is covered by particles at time T, assuming that the filter grain is covered by a monolayer of particles). This can be used to compare the contribution of deposited particles in the removal efficiency of deep bed filtration for suspensions with different physical and chemical characteristics.

Influence of particle size and size distribution in deep bed filtration and dynamic microfiltration

Particle size and size distribution is an important parameter in solid liquid separation process especially in granular bed filtration and in dynamic microfiltration. This paper discusses their effects on the above processes from extensive experimental data obtained. In granular bed filtration, the experimental results showed that the initial efficiency follows the pattern reported by previous experimental and theoretical studies, i.e., lower efficiency for particles which fall in the range of critical size of 1 m. However, the particle removal during the transient stage increased with an increase in particle size for the range of sizes studied. An attempt was made to quantify these effects in granular bed filtration using semi-empirical approach. In dynamic membrane filtration also, the particle size plays a major role in the retention. However, despite the relative thickness of the membrane (compared to particle size) dynamic microfiltration appears more as a sieving process; the retention is mainly related to the largest pore size.

Evidence that the bed nucleus of the stria terminalis contributes to the modulation of hypophysiotropic corticotropin-releasing factor cell responses to systemic interleukin-1β

Systemic infection activates the hypothalamic-pituitary-adrenal (HPA) axis, and brainstem catecholamine cells have been shown to contribute to this response. However, recent work also suggests an important role for the central amygdala (CeA). Because direct connections between the CeA and the hypothalamic apex of the HPA axis are minimal, the present study investigated whether the bed nucleus of the stria terminalis (BNST) might act as a relay between them. This was done by using an animal model of acute systemic infection involving intravascular delivery of the proinflammatory cytokine interleukin-1β (IL-1β, 1 μg/kg). Unilateral ibotenic acid lesions encompassing the ventral BNST significantly reduced both IL-1β-induced increases in Fos immunoreactivity in corticotropin-releasing factor (CRF) cells of the hypothalamic paraventricular nucleus (PVN) and corresponding increases in adrenocorticotropic hormone (ACTH) secretion. Similar lesions had no effect on CRF cell responses to physical restraint, suggesting that the effects of BNST lesions were not due to a nonspecific effect on stress responses. In further studies, we examined the functional connections between PVN, BNST, and CeA by combining retrograde tracing with mapping of IL-1β-induced increases in Fos in BNST and CeA cells. In the case of the BNST, these studies showed that systemic IL-1β administration recruits ventral BNST cells that project directly to the PVN. In the case of the CeA, the results obtained were consistent with an arrangement whereby lateral CeA cells recruited by systemic IL-1β could regulate the activity of medial CeA cells projecting directly to the BNST. In conclusion, the present findings are consistent with the hypothesis that the BNST acts as a relay between the CeA and PVN, thereby contributing to CeA modulation of hypophysiotropic CRF cell responses to systemic administration of IL-1β.

Medial prefrontal cortex control of the paraventricular hypothalamic nucleus response to psychological stress : possible role of the bed nucleus of the stria terminalis

The medial prefrontal cortex (mPFC) has been strongly implicated in control of the paraventricular nucleus of the hypothalamus (PVN) response to stress. Because of the paucity of direct projections from the mPFC to the PVN, we sought to investigate possible brain regions that might act as a relay between the two during psychological stress. Bilateral ibotenic acid lesions of the rat mPFC enhanced the number of Fos-immunoreactive cells seen in the PVN after exposure to the psychological stressor, air puff. Altered neuronal recruitment was seen in only one of the candidate relay populations examined, the ventral bed nucleus of the stria terminalis (vBNST). Furthermore, bilateral ibotenic acid lesions of the BNST caused a significant attenuation of the PVN response to air puff. To better characterize the structural relationships between the mPFC and PVN, retrograde tracing studies were conducted examining Fos expression in cells retrogradely labeled with cholera toxin b subunit (CTb) from the PVN and the BNST. Results obtained were consistent with an important role for both the mPFC and BNST in the mpPVN CRF cell response to air puff. We suggest a set of connections whereby a direct PVN projection from the ipsilateral vBNST is involved in the mpPVN response to air puff and this may, in turn, be modulated by an indirect projection from the mPFC to the BNST.

Comparisons of sidewall correction of bed shear stress in open-channel flows

When investigating sediment transport in laboratory open-channel flows, it is often necessary to remove sidewall effects for computing effective bed shear stress. Previous sidewall correction methods are subject to some assumptions that have not been completely verified, and different values of the bed shear stress may be obtained depending on the approach used in making sidewall corrections. This study provides a quantitative assessment of the existing correction procedures by comparing them to a new sidewall correction model proposed in this study. The latter was derived based on the shear stress function and equivalent roughness size for both rigid and mobile bed conditions, which were obtained directly from experimental measurements. The comparisons show that the Einstein correction formula and the Vanoni and Brooks method generally predict relatively lower and higher bed shear stresses, respectively, while the Williams’ empirical function leads to more scatter. This study also demonstrates that the widely used Vanoni and Brooks approach can be well approximated by a simple formula derived based on the Blasius resistance function. The sidewall effects, when removed in the different ways, would consequently affect the presentation of the bedload function. Experimental results of bedload transport, when plotted as the dimensionless transport rate against the dimensionless shear stress with the latter being corrected using the present model, exhibit less scatter than those associated with the previous procedures.