Unsteady-State Mass Transfer Models

A complete parametric study of the unsteady state mass transfer model clearly shows that tp, the pulse period, 4, the polarization, A, the aspect ratio, and DF, the duty factor have a profound effect on the evolution and the final shape of the deposit. Large polarization s and aspect ratios lead to deposition that is mass transfer controlled. This results in keyhole formation, as the concentration gradient inside a high aspect ratio trench is very large. On the other hand, when the deposition is kinetically controlled (i.e. for small values of polarization and aspect ratio) the gradient down the length of the trench is much smaller and deposition proceeds at nearly the bulk concentration. This leads to conformal deposition, as there is negligible variation in the deposition rate at the mouth and at the bottom of the trench. 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

Mathematical models derived from mass-conservation equations under unsteady-state conditions allow the calculation of the extracted mass at different bed locations, as a function of time. Semi-batch operation for the high-pressure gas is usually employed, so a fixed bed of solids is bathed with a flow of fluid. Mass-transfer models allow one to predict the effects of the following variables fluid velocity, pressure, temperature, gravity, particle size, degree of crushing, and bed-length. Therefore, they are extremely useful in simulation and design. 

An accurate design of unsteady-state catalytic processes requires knowledge about the catalyst behavior and reaction kinetics under unsteady-state conditions, unsteady-state mass and heat transfer processes in the catalyst particle and along the catalyst bed, and dynamic phenomena in the catalytic reactor. New approaches for reactor modeling and optimization become necessary. Together, these topics form a wide area of research that has been continuously developed since the 1960s. 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. 

The absorption of a gas during condensation of water vapor on a cold water droplet Is a complex process characterized by unsteady state mass and heat transfer [6]. In the classical development of absorption of a gas In a liquids three theoretical models have ensued The film theory, the penetration theory, and the boundary layer theory. Each model Invokes different assumptions which result in different conclusions. 

The developed dynamic reactor model for the simulation studies of the unsteady-state-operated trickle-flow reactor is based on an extended axial dispersion model to predict the overall reactor performance incorporating partial wetting. This heterogeneous model consists of unsteady-state mass and enthalpy balances of the reaction components within the gas, liquid and catalyst phase. The individual mass-transfer steps at a partially wetted catalyst particle are shown in Fig. 4.5. 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

The above model of settler flow behaviour, combined with entrainment backmixing was used by Aly (1972) to model the unsteady-state extraction of copper from aqueous solution, using Alamine 336 solvent. An identification procedure for the relevant flow parameters showed an excellent fit to the experimental data with very realistic entrainment backmixing factors, fL = fQ = 3.5 percent, the fraction of well-mixed flow in the settlers, (XX = ay = 5 percent and an overall mass transfer capacity coefficient, Ka = 25 s->. 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

As a rule, mathematical models of unsteady-state processes cannot be formulated by simple addition of time derivatives to the equations describing the steady-state behavior of the reaction system. They relate both to the reaction kinetics as well as to the heat and mass transfer processes. For example, modeling of unsteady-state processes in a fixed bed reactor requires accounting for the processes of heat and mass transfer between the catalyst surface and the bulk of gas phase, although for steady-state operation these factors can be neglected. 

Application of this procedure is illustrated by an example of analysis of unsteady-state processes in a single catalyst pellet [8, 9]. A separate consideration of this element allows estimation of the domains of parameters where certain stages of heat and mass transfer can be neglected and the mathematical model thus simplified. These criteria derived after assuming the steady-state reaction rate r are given in Tabic 2. 

During the extraction an unsteady process prevails. The present paper presents an unsteady state mathematical model for a fixed bed extractor (model 1). The overall mass transfer coefficients were calculated by matching the calculated and experimental values of oil loading in CO2. The results are compared with those obtained by the model developed by Catchpole et al, 1994 (model II). Good agreement between both models results and our experimental measurements were obtained, although the model II allows the best fit over the entire extraction curve. 

Unsteady-state mathematical model based on the advancing front model of Ho et al. [3] neglects external phase mass transfer resistance and the effect of membrane breakage and has no adjustable parameters. 

If the values of local mean bubble diameter and local gas flux are available, a fluid dynamic model can estimate the required influence of mass transfer and reactions on the fluid dynamics of bubble columns. Fortunately, for most reactions, conversion and selectivity do not depend on details of the inherently unsteady fluid dynamics of bubble column reactors. Despite the complex, unsteady fluid dynamics, conversion and selectivity attain sufficiently constant steady state values in most industrial operations of bubble column reactors. Accurate knowledge of fluid dynamics, which controls the local as well as global mixing, is however, essential to predict reactor performance with a sufficient degree of accuracy. Based on this, Bauer and Eigenberger (1999) proposed a multiscale approach, which is shown schematically in Fig. 9.13. 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

Our discussions on the film model for heat transfer showed an exact parallel with the corresponding mass transfer problem. The same parallel holds for unsteady-state transfer. Thus, for Fo = Xt/pC rl 0 (short contact time), the time-averaged heat transfer coefficient is given by... 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

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