Precipitation In Small Systems--II. Mean Field Equations More Effective Than Population Balance

Part I (Manjunath et al., 1994, Chem. Engng Sci. 49, 1451-1463) of this paper showed that the random particle numbers and size distributions in precipitation processes in very small drops obtained by stochastic simulation techniques deviate substantially from the predictions of conventional population balance. The foregoing problem is considered in this paper in terms of a mean field approximation obtained by applying a first-order closure to an unclosed set of mean field equations presented in Part I. The mean field approximation consists of two mutually coupled partial differential equations featuring (i) the probability distribution for residual supersaturation and (ii) the mean number density of particles for each size and supersaturation from which all average properties and fluctuations can be calculated. The mean field equations have been solved by finite difference methods for (i) crystallization and (ii) precipitation of a metal hydroxide both occurring in a single drop of specified initial supersaturation. The results for the average number of particles, average residual supersaturation, the average size distribution, and fluctuations about the average values have been compared with those obtained by stochastic simulation techniques and by population balance. This comparison shows that the mean field predictions are substantially superior to those of population balance as judged by the close proximity of results from the former to those from stochastic simulations. The agreement is excellent for broad initial supersaturations at short times but deteriorates progressively at larger times. For steep initial supersaturation distributions, predictions of the mean field theory are not satisfactory thus calling for higher-order approximations. The merit of the mean field approximation over stochastic simulation lies in its potential to reduce expensive computation times involved in simulation. More effective computational techniques could not only enhance this advantage of the mean field approximation but also make it possible to use higher-order approximations eliminating the constraints under which the stochastic dynamics of the process can be predicted accurately.

Bridged systems via radical cyclization reactions. Stereospecific synthesis of chiral tricyclo[6.2.1.0(1,5)]undecanes

The application of radical-mediated cyclizations and annulations in organic synthesis has grown in importance steadily over the years to reach the present status where they are now routinely used in the strategy-level planning.2 The presence of a quaternary carbon atom is frequently encountered in terpenoid natural products, and it often creates a synthetic challenge when two or more quaternary carbon atoms are present in a contiguous manner.3 Even though creation of a quaternary carbon atom by employing a tertiary radical is very facile, creation of a quaternary carbon atom (or a spiro carbon atom) via radical addition onto a fully substituted olefinic carbon atom is not that common but of synthetic importance. For example, the primary radical derived from the bromide 1 failed to cyclize to generate the two vicinal quaternary carbon atoms and resulted in only the reduced product 2.4 The tricyclic carbon framework tricyclo[6.2.1.01,5]undecane (3) is present in a number of sesquiterpenoids e.g. zizzanes, prelacinanes, etc.5

Ionic transport in (PEG)x LiBr systems

Ionic conductivity in (PEG)(x)LiBr systems is measured using the complex impedance method in the temperature range -20 degrees C to 100 degrees C. For x = 6 and 10, above a certain concentration dependent temperature T-c, a power law fit based on mode coupling theory is seen to better explain the data than the Vogel-Tamman-Fulcher (VTF) expression. Li-7 NMR linewidth measurements indicate two regions of motional narrowing, one attributable to segmental motion and the other to translational diffusion.

Approximate Controllability Of Semilinear Systems Using Integral Contractors

In this article, a non-autonomous (time-varying) semilinear system is considered and its approximate controllability is investigated. The notion of 'bounded integral contractor', introduced by Altman, has been exploited to obtain sufficient conditions for approximate controllability. This condition is weaker than Lipschitz condition. The main theorems of Naito [11, 12] are obtained as corollaries of our main results. An example is also given to show how our results weaken the conditions assumed by Sukavanam[17].