Equilibrium and kinetics of water adsorption in carbon molecular sieve: theory and experiment

Measurements of water adsorption equilibrium and kinetics in Takeda carbon molecular sieve (CMS) were undertaken in an effort to characterize fundamental mechanisms of adsorption and transport. Adsorption equilibrium revealed a type III isotherm that was characterized by cooperative multimolecular sorption theory. Water adsorption was found to be reversible and did not display hysteresis upon desorption over the conditions studied. Adsorption kinetics measurements revealed that a Fickian diffusion mechanism governed the uptake of water and that the rate of adsorption decreased with increasing relative pressure. Previous investigations have attributed the observed decreasing trend in the rate of adsorption to blocking of micropores. Here, it is proposed that the decrease is attributed to the thermodynamic correction to Fick's law which is formulated on the basis of the chemical potential as the driving force for transport. The thermodynamically corrected formulation accounted for observations of transport of water and other molecules in CMS.     

Water sorption in silicone foam containing diatomaceous earth

Room temperature vulcanizing (RTV) silicone foams are commonly used for compression sealing, structural support, packaging, and damping applications. The presence of sorbed water in foams can affect the mechanical and chemical properties of these materials. In order to investigate water sorption behavior, a silicone foam containing diatomaceous earth filler was synthesized and studied for water uptake characteristics at 20, 50, and 80 degrees C. Type II equilibrium and bimodal kinetic behavior that was governed by a rapid initial uptake followed by a prolonged sorption over a larger time scale was observed. In order to explain this bimodal behavior, the major components of this foam-the silicone polymer and the diatomaceous earth-were independently studied for their water equilibrium behavior and uptake kinetic characteristics. Type II equilibrium was observed for both components. The kinetic behavior of the silicone polymer was governed by a very rapid uptake of water. The kinetic behavior of the diatomaceous earth was governed by a rapid initial uptake followed by a prolonged sorption over a larger time scale. A physically based and thermodynamically consistent mathematical model describing the water equilibrium and kinetics in diatomaceous earth and silicone polymer components, was employed to characterize the data. This model formed the basis of a predictive model for estimation of water sorption in filled silicone foam. The predictive model was tested against sorption and desorption data yielding favorable results for a range of temperatures.     

Apparate zur Gasanalyse

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A Bayesian analysis of the compression set and stress-strain behavior in a thermally aged silicone foam

Data obtained from an archived nine-year aging study on S5370 foam were used to develop compression set and stress-strain aging models. Compression set was characterized using a first order kinetic model and the stress-strain relationship was analyzed using a material model previously described by Rusch for flexible foams. The models were fitted to data from the aging study using Bayesian methods, which easily accommodate uncertainties in the test conditions and provide probability distributions of the model parameters. The parameter distributions were sampled using a Markov chain Monte Carlo algorithm and incorporated to effect prediction intervals and compared to data obtained from independent studies for the purpose of validation. Compression set data from the short time study of Patel and Skinner are shown to predict significantly higher compression sets, which are attributed to additional crosslinking reactions and other phenomena that do not dominate the long term aging behavior. Using data from the nine-year study, the time period required to achieve a given compression set at 25 °C is increased by 20 years or more over the predictions of Patel and Skinner. The activation energy applicable near room temperature is similar to that reported by Patel and Skinner, which is consistent with numerous physical and catalyzed chemical mechanisms. Finally, load retention predictions from the stress-strain aging model agree with independent studies at test gaps that are larger than or equal to a zero gradient test gap limit.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.