Diffusion-fluid flow model

Above-mentioned reaction, diffusion and advection influence mass transfer in rock-water system. It is generally difficult to solve the differential equation including all these mechanisms. Thus, the two coupled models at constant temperature and pressure will be explained below. They are (1) reaction-fluid flow model, (2) reaction-diffusion model, (3) diffusion-fluid flow model. In addition to these coupled models, model taking into account the change in temperature will be considered. 

Step 1 It is easy to add the convective diffusion equation by starting with the fluid flow model of the T-sensor. Choose Multiphysics/Model Navigator. A window appears with a hst of possible equations. Scroll down and select Chemical Engineering Module/Mass Balance/Convection and Diffusion/Steady-state Analysis. Click Add. Now FEMLAB will solve both equations. 

Clearly there is no theoretical limit to the complexity of the reactions that might be considered in this way. In addition, it is quite possible to couple this type of calculation with other types, such as fluid flow, heat flow, pressure changes, diffusion, permeability changes, deformation, and so on, because these other model calculations also are carried out iteratively in a large series of small steps. Thus, for example, after carrying out a Af step in a reaction path model, we could then take a small step in a heat flow model, then a small step in a fluid flow model, and then return to the reaction model, and so on. The heat... 

In this book we considered mass transfer and elemental migration between the atmosphere, hydrosphere, soils, rocks, biosphere and humans in earth s surface environment on the basis of earth system sciences. In Chaps. 2, 3, and 4, fundamental theories (thermodynamics, kinetics, coupling model such as dissolution kinetics-fluid flow modeling, etc.) of mass transfer mechanisms (dissolution, precipitation, diffusion, fluid flow) in water-rock interaction of elements in chemical weathering, formation of hydrothermal ore deposits, hydrothermal alteration, formation of ground water quality, seawater chemistry. However, more complicated geochemical models (multi-components, multi-phases coupled reaction-fluid flow-diffusion model) and phenomenon (autocatalysis, chemical oscillation, etc.) are not considered. 

Chemical equilibrium and mass transfer mechanisms (chemical reactions, diffusion, fluid flow (advection), adsorption, etc.) (Chaps. 1,2, and 3) are examined in order to illustrate the compositional variation that exists within water (ground water, hydrothermal solution, seawater) and weathered and hydrothermally altered rocks and soils. To better understand the subsystems of the earth, equilibrium and mass transfer coupling models are apphed to the seawater system, as an example of a low-temperature exogenic system, and hydrothermal systems, as an example of high-temperature endogenic systems (Chap. 4). 

GASFLOW models geometrically complex containments, buildings, and ventilation systems with multiple compartments and internal structures. It calculates gas and aerosol behavior of low-speed buoyancy driven flows, diffusion-dominated flows, and turbulent flows dunng deflagrations. It models condensation in the bulk fluid regions heat transfer to wall and internal stmetures by convection, radiation, and condensation chemical kinetics of combustion of hydrogen or hydrocarbon.s fluid turbulence and the transport, deposition, and entrainment of discrete particles. 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimension. Secondly, two-dimensional dynamics permits easier (sometimes direct) comparison to real physical systems. As we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Consider the scaleup of a small, tubular reactor in which diffusion of both mass and heat is important. As a practical matter, the same fluid, the same inlet temperature, and the same mean residence time will be used in the small and large reactors. Substitute fluids and cold-flow models are sometimes used to study the fluid mechanics of a reactor, but not the kinetics of the reaction. 

The flux vector accounts for mass transport by both convection (i.e., blood flow, interstitial fluid flow) and conduction (i.e., molecular diffusion), whereas S describes membrane transport between adjacent compartments and irreversible elimination processes. For the three-subcompartment organ model presented in Figure 2, with concentration both space- and time-dependent, the conservation equations are... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

The diffusion process in general may be viewed as the model for specific well-defined transport problems. In particle diffusion, one is concerned with the transport of particles through systems of particles in a direction perpendicular to surfaces of constant concentration in a viscous fluid flow, with the transport of momentum by particles in a direction perpendicular to the flow and in electrical conductivity, with the transport of charges by particles in a direction perpendicular to equal-potential surfaces. 

Numerous modeling studies of CVD reactors have been made and are summarized in recent review papers (I, 212). Table 3 in reference 212 lists major examples of CVD models up to mid-1986. Therefore, rather than giving an exhaustive list of previous work, Table V presents a summary of the major modeling approaches and forms the basis for the ensuing discussion, which is most appropriately handled in terms of two groups (1) hot-wall LPCVD systems and (2) cold-wall, near-atmospheric-pressure reactors. In LPCVD reactors, diffusion and surface reaction effects dominate, whereas in cold-wall reactors operated at near-atmospheric pressures, fluid flow and gas-phase reactions are important in predicting performance, as discussed earlier in relation to transport phenomena. 

LPCVD reactor modeling involves many of the same issues of multi-component diffusion reactions that have been studied in the past decade in connection with heterogeneous catalysis. Complex fluid-flow phenomena strongly affect the performance of atmospheric-pressure CVD reactors. Two-dimensional and some three-dimensional flow structures in the classical horizontal and vertical CVD reactors have been explored through flow visual-... 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

Xu and Ruppel (1999) solved the coupled mass, heat, and momentum equations of change, for methane and methane-saturated fluxes from below into the hydrate stability region. They show that frequently methane is the critical, limiting factor for hydrate formation in the ocean. That is, the pressure-temperature envelope of the Section 7.4.1 only represents an outer bound of where hydrates might occur, and the hydrate occurrence is usually less, controlled by methane availability as shown in Section 7.4.3. Further their model indicates the fluid flow (called advection or convection) in the amount of approximately 1.5 mm/yr (rather than diffusion alone) is necessary to produce significant amount of oceanic hydrates. 

Good quality RO membranes can reject >95-99% of the NaCl from aqueous feed streams (Baker, Cussler, Eykamp et al., 1991 Scott, 1981). The morphologies of these membranes are typically asymmetric with a thin highly selective polymer layer on top of an open support structure. Two rather different approaches have been used to describe the transport processes in such membranes the solution-diffusion (Merten, 1966) and surface force capillary flow model (Matsuura and Sourirajan, 1981). In the solution-diffusion model, the solute moves within the essentially homogeneously solvent swollen polymer matrix. The solute has a mobility that is dependent upon the free volume of the solvent, solute, and polymer. In the capillary pore diffusion model, it is assumed that separation occurs due to surface and fluid transport phenomena within an actual nanopore. The pore surface is seen as promoting preferential sorption of the solvent and repulsion of the solutes. The model envisions a more or less pure solvent layer on the pore walls that is forced through the membrane capillary pores under pressure. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

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