A Simple Markovian Model for the Initial Stage of the Formation of Carbon Molecular Sieves by Carbon Deposition

Abstract

One type of molecular sieves is carbon molecular sieves (CMSs). Activated carbons, with pores of relatively uniform pore-size distribution, which can be produced from various sources including biomass, are suitable as substrates to manufacture CMSs. This is accomplished by narrowing the pores' mouths of such activated carbons through deposition of fine particles of carbon onto them. These carbon particles are generated by decomposing a gaseous carbon source. The formation of CMSs proceeds stochastically, which is mainly attributable to mesoscopic size and complex motion of the depositing carbon particles and random distribution of the pores on the activated carbons. Herein, CMS formation is modeled as a Markovian process, specifically, a pure-birth process with a constant intensity of transition. Such a pure-birth process can possibly lead to the simplest stochastic model for the kinetics of CMS formation, especially at the initial stage. In spite of its simplicity, this model is useful for preliminary exploration of stochastic characteristics of CMS formation. The model gives rise to the master equation of the pure-birth process as well as the governing equations for the mean and variance of the random variable characterizing CMS formation. To validate the model, the analytical solutions of these equations have been compared with available experimental data; the model is in excellent accord with the data. Meanwhile, the analysis of these available data through stochastic modeling has revealed that they are highly accurate.