Mathematical modeling physical-mass transfer models

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

In the model described in this work every effort has been made to ensure that it embodies physically meaningful parameters. It is inevitcible, however, that some simplistic idealizations of the physical processes involved in GPC must be made in order to arrive at a system of equations which lends itself to mathematical solution. The parameters considered are, the axial dispersion, interstitial volume fraction, flow rate, gel particle size, column length, intra-particle diffusivity, accessible pore volume fraction and mass transfer between the bulk solution eind the gel particles. 

This involves knowledge of chemistry, by the factors distinguishing the micro-kinetics of chemical reactions and macro-kinetics used to describe the physical transport phenomena. The complexity of the chemical system and insufficient knowledge of the details requires that reactions are lumped, and kinetics expressed with the aid of empirical rate constants. Physical effects in chemical reactors are difficult to eliminate from the chemical rate processes. Non-uniformities in the velocity, and temperature profiles, with interphase, intraparticle heat, and mass transfer tend to distort the kinetic data. These make the analyses and scale-up of a reactor more difficult. Reaction rate data obtained from laboratory studies without a proper account of the physical effects can produce erroneous rate expressions. Here, chemical reactor flow models using mathematical expressions show how physical... 

The properties of wood(7,14) were used to analyze time scales of physical and chemical processes during wood pyrolysis as done in Russel, et al (15) for coal. Even at combustion level heat fluxes, intraparticle heat transfer is one to two orders of magnitude slower than mass transfer (volatiles outflow) or chemical reaction. A mathematical model reflecting these facts is briefly presented here and detailed elsewhere(16). It predicts volatiles release rate and composition as a function of particle physical properties, and simulates the experiments described herein in order to determine adequate kinetic models for individual product formation rates. 

In this case, as shown in Figure 4, the subsystems are stoichiometry, material balance, energy balance, chemical kinetics, and interphase mass transfer. The mass transfer phenomena can be subdivided into (1) phase equilibrium which defines the driving force and (2) the transport model. In a general problem, chemical kinetics may be subdivided into (1) the rate process and (2) the chemical equilibrium. The next step is to develop models to describe the subsystems. Except for chemical kinetics, generally applicable mathematical equations based on fundamental principles of physics and chemistry are available for describing the subsystems. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

In the Multigrain model, fractured catalyst microparticles are produced during the polymerization and uniformly dispersed in the polymer each of these particles behaves as a micro Solid core and diffusion within them, as well as in the interstices between them, can take place. In the Polymeric flow model the catalyst microparticles are dispersed in a polymer continuum and move outward in proportion to the volumetric expansion due to polymerization only one value of diffusivity is considered. Both these models predict significant MWD broadening due to mass transfer limitations (Q , 9 for polypropylene in the Polymeric flow model) on the basis of mathematical calculations carried out assuming reasonable values of the kinetic and physical parameters. 

The physical aspects involved are heat transfer and mass transfer. The external heat and mass transfers are free of resistance under the environment of fluidized beds. Only the interna heat and mass transfers are treated here. To develop the mathematic model, the following assumptions and statements are made ... 

As described in the subsequent chapters in Part m, models for the impedance response can be developed from proposed hypotheses involving reaction sequences (e.g., Chapters 10 and 12), mass transfer (e.g., Chapters 11 and 15), and physical phenomena (e.g.. Chapters 13 and 14). These models can often be expressed in the mathematical formalism of electrical circuits. Electrical circuits can also be used to construct a framework for accounting for the phenomena that influence the impedance response of electrochemical systems. A method for using electrical circuits is presented in this chapter. 

Conceptual models of percutaneous absorption which are rigidly adherent to general solutions of Pick s equation are not always applicable to in vivo conditions, primarily because such models may not always be physiologically relevant. Linear kinetic models describing percutaneous absorption in terms of mathematical compartments that have approximate physical or anatomical correlates have been proposed. In these models, the various relevant events, including cutaneous metabolism, considered to be important in the overall process of skin absorption are characterized by first-order rate constants. The rate constants associated with diffusional events in the skin are assumed to be proportional to mass transfer parameters. Constants associated with the systemic distribution and elimination processes are estimated from pharmacokinetic parameters derived from plasma concentration-time profiles obtained following intravenous administration of the penetrant. 

Successful approaches to designing an extraction process begin with an appreciation of the fundamentals (basic phase equilibrium and mass-transfer principles) and generally rely on both experimental studies and mathematical models or simulations to define the commercial technology. Small-scale e3q)eriments using representative feed usually are needed to accurately quantify physical properties and phase equilibrium. Additionally, it is common practice in industry to perform... 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

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. 

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