(422b) A Non-Linear Stochastic Model for the Formation of Activated Carbons
Among carbon adsorbents, activated carbons (ACs) are exceedingly effective for the purification of gases and liquids or the separation of their mixtures. In the formation of ACs, the original internal surfaces of carbonaceous substrates, e.g., coal or biomass, are modified via a variety of chemical and/or physical methods, thereby augmenting the carbonaceous substrates' adsorbing capacities. The formation of ACs tends to proceed randomly or stochastically in view of the discrete and mesoscopic nature of the carbonaceous substrates as well as the random encounters between the activation agent and carbon on the carbonaceous substrates' internal surfaces; moreover, the carbonaceous substrates' internal surfaces exhibit an intricate morphology, or structure. Naturally, these traits of the formation of ACs render the process to vary incessantly with time. Thus, it is highly desirable that the analysis, modeling, and simulation of the formation of ACs from carbonaceous substrates be performed in light of a stochastic paradigm. Herein, a stochastic model for the formation of ACs is formulated as a pure-birth process based on a highly non-linear intensity of transition, which incorporates the effects of morphological, i.e., structural, characteristics of the carbonaceous substrates' internal surfaces. The model gives rise to the process' non-linear master equation whose solution is obtained by resorting to a rational approximation method, the system-size expansion. This solution renders it possible to compute the mean as well as higher moments about this mean, e.g., variance or standard deviation, which are useful to quantify the process' inherent fluctuations. Furthermore, the master equation is simulated via the Monte Carlo method at the initial stage of the formation of ACs where the process' fluctuations tend to magnify. The results of non-linear modeling are validated by comparing them with the available experimental data as well as with the results from Monte Carlo simulation; they are also compared with the results obtained from a linear stochastic model for the formation of ACs.