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The Polymeric Matrix and its Influence
Published in Maik W. Jornitz, Theodore H. Meltzer, Sterile Filtration, 2020
Maik W. Jornitz, Theodore H. Meltzer
In keeping with the Langmuir adsorption isotherm, the adsorption rate is highest at the inception of the filtration. The initial effluent consequently contains the least protein. Progressive saturation of the filter causes a decrease in the adsorption rate and an increase in the effluent’s protein concentration. At the filter’s saturation point, no further adsorption takes place, and the effluent’s concentration approaches that of the feedstream. The defining of the filter’s saturation curve is essential to the filtration process design (Sundaram, 1998). In most cases involving microporous membranes a steady-state situation occurs, with the filter adsorbing only a comparatively small amount of protein. Exceptions to the attainment in the effluent of the protein concentration present in the feedstream can be due to losses caused by the sieve removal of undissolved or gel-associated protein. These happenings reflect the protein “particle” size and the pore size of the filter. When an ultrafilter is used, the molecular weight cutoff, the dalton rating of the filter, is a factor. The pore size and hence the pore size distribution (although the latter is seldom known or measured) influence the filter’s adsorptive performance. Flow varies as the fourth power of the pore radius. Therefore, per unit time more solution flows through a microporous membrane than through the smaller pore ultrafilter. This conveys more protein to be absorbed per unit time. However, the more open pores of the microporous membrane cause less protein to be adsorbed initially, more of the solution passing through the filter without encountering the pore walls. On a time basis, however, the greater volume of flow through the microporous membrane compensates; protein retention increases over time. It was shown by Devereaux and Hoare (1985) that flux declined more rapidly for a 0.2 μm rated membrane than for an ultrafilter of 50,000 Da given the same feedstream.
Using Laboratory Data to Design a Full-Scale Treatment System
Published in Evan K. Nyer, Practical Techniques for Groundwater and Soil Remediation, 2019
If it is necessary to remove a certain size or type of suspended solid in a treatment system, then a sand filter is the wrong unit of operation. In those circumstances the designer should rely upon a different type of filter, i.e., cartridge filter, ultrafilter, etc.
A dynamical proof of the van der Corput inequality
Published in Dynamical Systems, 2022
Nikolai Edeko, Henrik Kreidler, Rainer Nagel
Let and be a nonprincipal ultrafilter such that Denote by the canonical extension of to and for each let be the Dirac measure at n. Then Here, the probability measure exists as a weak*-limit because any limit along an ultrafilter exists in a compact space. Now, let be the Koopman operator for the right shift and let denote its mean ergodic projection. By the Cauchy–Schwarz inequality and since is a conditional expectation, we conclude that
Synthesis and characterization of stable lead (II) orthophosphate nanoparticle suspensions
Published in Journal of Environmental Science and Health, Part A, 2020
Darren A. Lytle, Casey Formal, Evelyne Doré, Christy Muhlen, Stephen Harmon, Daniel Williams, Simoni Triantafyllidou, Maily Pham
Additional stable Pb nanoparticle challenge water suspensions were prepared in triplicate from primary soluble Pb solutions as described. Water analysis results were reproducible across runs and consistent with time (up to 24 hours) and, therefore, were analyzed as sets. The average pH of the suspensions was 7.57 ± 0.10 and the average temperature was 24.1 ± 0.3 °C. Total Pb concentration averaged 0.097 ± 0.010 mg/L. The Pb passing the 0.2 µm filter averaged 0.037 ± 0.012 mg/L (approximately 40% of total Pb) and the Pb passing the ultrafilter (soluble Pb) averaged 0.001 ± 0.000 mg/L (1% of total Pb). The fraction of Pb passing the 0.2 µm was relatively variable but did not trend with time. For example, 0.2 µm Pb values ranged between 0.013 and 0.060 mg/L (Figure 4). Soluble Pb did not change with dilution of Pb-phosphate nanoparticles (i.e., nanoparticles did not dissolve in the challenge water). Total phosphate concentration of stable Pb nanoparticle challenge water averaged 3.7 ± 0.3 mg PO4/L. The total phosphate concentration passing the 0.2 µm averaged 3.7 ± 0.3 mg PO4/L (99.7% of total phosphate) and the total phosphate passing the ultrafilter (soluble phosphate) averaged 3.7 ± 0.3 mg PO4/L (99.3% of total phosphate).
Two proofs of the algebraic completeness theorem for multilattice logic
Published in Journal of Applied Non-Classical Logics, 2019
Oleg Grigoriev, Yaroslav Petrukhin
A pair is called an ultralogicaln-lattice (ultralogical multilattice) iff is an n-lattice and satisfies the following conditions, for each , , and : iff and ( is an n-filter (multifilter) on ); iff or ( is a prime n-filter on ); iff ( is an n-ultrafilter (ultramultifilter) on ).