S.T. Chou

Stochastic modeling of transient residence-time distributions during start-up

Abstract The theory of residence-time distribution, RTD theory in short, is a cornerstone of chemical engineering science and practice, in general, and that of chemical reactor analysis and design, in particular. The creation of the modern, systematic RTD theory has been attributed to Danckwerts. As evident from his liberal adoption of terminologies of probability and statistics, he was apparently well aware of the stochastic nature of the process that gives rise to a residence-time distribution. While Danckwerts steered the development of the RTD theory essentially along the path of deterministic physics, obviously, the description of RTD is better couched in the statistical or stochastic parlance. Stochastic modeling visualizes the fluid in a flow system as being composed of discrete entities. This visualization reveals a greater insight into the underlying mechanism than deterministic modeling, thereby facilitating our understanding of the flow and mixing characteristic of the system. In the present work, an attempt has been made to derive a unified mathematical model of the RTD during process start-up by rigorously resorting to the theories and methodologies of stochastic processes. Specifically, the expressions for RTDs of molecules, fluid particles or any flowing entities passing through continuous flow systems have been derived from the stochastic population balance of these molecules, particles or entities. The resultant expressions are applicable to both unsteady-state and steady-state flow conditions.

Stochastic analysis of stepwise cellulose degradation

Abstract Cellulose is a linear polymer of glucose. The enzymatic degradation of cellulose can be stepwise or random depending on the enzymes acting on it. In this work, a stochastic approach is used to model stepwise degradation. The model predicts the distribution, mean and variance of the number or concentration of cellulose chains with j glucose units arising from hydrolysis at any time t . Estimates of the intensities (or reaction rate constants) of degradation are obtained by fitting the model to experimental data.

Modelling and simulation of deep-bed filtration: a stochastic compartmental model

Abstract To gain insight into the performance of a deep-bed filter, it is essential to determine the spatial distribution of suspended particles in the bed as a function of time. More often than not, a filtration process behaves stochastically rather than deterministically; therefore, a stochastic compartmental model is proposed to simulate the concentration dynamics of suspended particles in the liquid and solid parts over the different sections of the filter. In this study, the filter bed is divided into an arbitrary number of compartments in the direction of flow. The model yields the distribution of suspended particles along the bed at any time t , including those exited from the bed up to that time, and the mean and variance of the distribution. The parameters of the model are estimated by fitting the model to the experimental data. The reasonably good agreement between the simulated results and experimental data ensures the applicability of the present model.

BIRTH-DEATH MODELING OF DEEP BED FILTRATION: SECTIONAL ANALYSIS

Stochastic models, namely, the second-order, pure birth process and the linear, birth-death process, have been employed in conjunction with the Carman-Kozeny equation to simulate the performance of the deep-bed filter in terms of the pressure drop dynamics under a constant flow condition. These models take into account the blockage of pores by suspended particles and/or scouring of deposited particles; the filter bed is assumed to be composed of several compartments. The present models appear to represent the majority of the available experimental data.

Modeling of complex chemical reactions in a continuous-flow reactor: a Markov chain approach

Abstract To simulate the dynamics of a chemically reacting system as a Markov chain, the states of the chain need to be properly identified. In the present approach, a molecule is viewed as an “object”, “entity” or system. The transformation of the molecule from one species to another is visualized as the transition of this “entity” from one form to another. Furthermore, appropriately selected chemical species or entities in the mixture serve as the states of the chain. The selection of such species is subject to the stoichiometric constraint based on the atomic balance. The approach is illustrated with several chemically reacting systems. The results are in good agreement with the known results obtained from the deterministic approach.

An experimental study of deep-bed filtration: stochastic analysis

Abstract A series of experimental runs of deep-bed filtration was conducted with suspensions of solids of various sizes, size distributions and concentrations. For each run, the pressure drop and effluent concentration were measured as functions of filtration time. The stochastic models, the linear, birth—death, the linear, pure birth, and the second-order, pure birth models, show a good fit with the pressure drop data; the difference between one model and another depends mainly on the relative sizes of coal particles and sand.

The surface-renewal theory of interphase transport : a stochastic treatment

Abstract The stagnant-film, boundary-layer, and surface-renewal theories have been regarded as the cornerstone of the science of interphase mass or heat transfer in turbulent environments. The stagnant-film theory has been highly popular and remains so because of its simplicity; however, it is deemed too simplistic and unrealistic. The boundary-layer theory has been derived from a fairly rigorous and self-consistent fluid mechanical theory based on the notion of continuity; nevertheless, this theory is incapable of elucidating random disturbance or chaotic bursting at the interface under turbulent conditions. The surface-renewal theory has been conceived so that the deficiencies of the first two theories can be rectified through incorporation of some statistical components into the description of interphase mass or heat transport. Numerous variants of this theory, giving rise to various mathematical models, have been proposed; still, the acceptance or popularity of the surface-renewal theory appears to lag behind the other two theories. This is probably attributable to the fact that while the mathematical formulation of the theory is abundantly couched in statistical or stochastic parlance, the methodologies and procedures followed are those of continuum mechanics and deterministic mathematics, thereby rendering the model or theory less acceptable. The present work attempts to derive a self-consistent mathematical model of the surface-renewal theory of interphase mass transfer by resorting to the theories and methodologies of stochastic processes based on the Markovian assumption. Specifically, the expression for the contact-time distribution of fluid elements or solid particles participating in the interphase transport has been derived from the stochastic population balance of these elements or particles. Moreover, the expression for the dynamic rate of transfer of molecules or microscopic components across the interface has been derived as the continuous limit of the probability balance equation of the random walks of these entities around the interface. Proper coupling of the two expressions constitutes the desired model. By analogy, this model is applicable to the turbulent interphase heat transport and plausibly to the momentum transport under certain circumstances.

Transient analysis of crystallization: effect of the initial size distribution

The effect of seed size distribution on the transient and steady-state crystal size distribution in a continuous mixed suspension, mixed product removal crystallizer has been analyzed and simulated from a stochastic point of view. The resultant stochastic model provides an easier way of obtaining the dynamic information on the performance of the crystallizer under consideration than the conventional population balance techniques. Moreover, the moments of the number of crystals at each size can be evaluated. Simulation results indicate that the seed size distribution gives rise to a maximum in the crystal size distribution. The steady-state representation can be recovered from the resultant dynamic model; it reduces to the conventional deterministic population balance model under the condition of no random variation.

THE MASTER EQUATION FOR LINEAR ADSORPTION AND DESORPTION OF GASES ON SOLID SURFACES

A Markovian model has been derived for a process involving reversible physisorption and irreversible chemical adsorption of simple gaseous molecules on a solid surface, which obey linear rate laws. The model is written in terms of the conditional probability of transition between two populations, physisorbed and chemisorbed molecules. The resultant expression, the master equation of the process, has given rise to the governing differential equations for the mean and variance of the coverage of the solid surface by the gaseous molecules; these equations have been solved analytically. The variance or coefficient of variation, expressing the magnitude of fluctuations, is substantial for a small-size system, e.g., a highly evacuated and/or dilute system. It is not uncommon to find such a system at the commencement and conclusion of any process; these periods are the most critical from the standpoint of operation, monitoring and control.

Modeling Fluctuations in the Growth Rate of a Single Crystal

The growth rate of a single crystal depends on various mesoscopic variables, e.g., irregularities such as defects and steps in the surface. It is highly plausible that such irregularities render the growth rates of crystals of an identical size to vary randomly. This work aims at modeling stochastically the fluctuations in the growth rate of a single crystal in a crystallizer. The crystal size in any of the equally-divided domains is considered as the random variable defining the state of the system. The transition of the crystal size from one state to another is characterized by a set of transition-intensity functions which, as for the case of the deterministic growth rate, exhibit a power-law dependence on the size. The master equation for the system is formulated through probabilistic balance around a particular state by taking into account all mutually exclusive events. The resultant nonlinear master equation has been expanded in power series of a small parameter, i.e., the reciprocal of the maximum crystal size obtainable, by means of the system-size expansion. This has yielded expressions for the means and variances of the size of a single particle. Comparison of the simulated crystal size with available experimental data indicates that the present stochastic model adequately portrays the dynamic behavior of a single crystal.