Diffusive mixing Dynamical equation

Great care has to be given to the physics of rotation and to the treatment of its interaction with mass loss. For differentially rotating stars, the structure equations need to be written differently [9] than for solid body rotation. For the transport of the chemical elements and angular momentum, we consider the effects of shear mixing, meridional circulation, horizontal turbulence and in the advanced stages the dynamical shear is also included. Caution has to be given that advection and diffusion are not the same physical effect. 

Of all of the methods reviewed thus far in this book, only DNS and the linear-eddy model require no closure for the molecular-diffusion term or the chemical source term in the scalar transport equation. However, we have seen that both methods are computationally expensive for three-dimensional inhomogeneous flows of practical interest. For all of the other methods, closures are needed for either scalar mixing or the chemical source term. For example, classical micromixing models treat chemical reactions exactly, but the fluid dynamics are overly simplified. The extension to multi-scalar presumed PDFs comes the closest to providing a flexible model for inhomogeneous turbulent reacting flows. Nevertheless, the presumed form of the joint scalar PDF in terms of a finite collection of delta functions may be inadequate for complex chemistry. The next step - computing the shape of the joint scalar PDF from its transport equation - comprises transported PDF methods and is discussed in detail in the next chapter. Some of the properties of transported PDF methods are listed here. 

Heat and mass transfer constitute fundamentally important transport properties for design of a fluidized catalyst bed. Intense mixing of emulsion phase with a large heat capacity results in uniform temperature at a level determined by the balance between the rates of heat generation from reaction and heat removal through wall heat transfer, and by the heat capacity of feed gas. However, thermal stability of the dilute phase depends also on the heat-diffusive power of the phase (Section IX). The mechanism by which a reactant gas is transferred from the bubble phase to the emulsion phase is part of the basic information needed to formulate the design equation for the bed (Sections VII-IX). These properties are closely related to the flow behavior of the bed (Sections II-V) and to the bubble dynamics. 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

Within this interacting-filament regime of the fast binary reaction under mixing by a complex flow, the approach of the product amount to its final equilibrium value is expected to proceed exponentially. This can be argued by considering the equations for product width and interface effective dimension (Karolyi and Tel, 2005) or by noting from Eqs. (5.8)-(5.9) that the decay of the product concentration C towards its final value T/2 is described by the dynamics of the auxiliary field [Pg.162]

We now discuss some of the main features of LLPTC models developed for reaction under neutral conditions. Evans and Palmer (1981) were among the first to consider the effect of diffusion and mass transfer inPTC. They considered PTC in liquid-liquid systems by considering two well-mixed bulk phases of uniform composition separated by a uniform stagnant mass-transfer layer at the interface, and set up equations for bulk phase species balance and mass conservation equations for simultaneous diffusion and reaction in the film. Dynamics of the interaction between reaction and diffusion were studied under these assumptions for two special cases (a) reaction which is pseudo-first-order in the quaternary ion-pair (b) mass-transfer controlled instantaneous reaction. 

One natural approach to describing mixing is to solve the equations of motion of the fluid. In fluid systems, the type of fluid flow is obviously important, and we should consider both laminar and turbulent flow, and various mechanisms of diffusion (molecular diffusion, eddy diffusion). Using fluid mechanics to describe all cases of interest is a difficult problem, both from the modeling and computational perspectives. Rapid developments in computational fluid dynamics (CFD), however, make this approach increasingly attractive 1]. 

Biot and Darcy theory shortcomings have been largely overcome by development of a coupled diffusion-dynamic formalism (de la Cruz et al. 1993, Spanos 2001, Spanos et al. 2(X)3). Porosity is treated as an explicit thermodynamic variable, so that dnumerical model development. Nevertheless, if they are solved subject to the assumption of the incompressibility of a liquid saturant, the existence of a slow wave is predicted. It is called the porosity dilation (PD) wave it is not a strain wav, it is a coupled liquid-solid displacement wave, and it has some interesting properties. 

For a modelling of adsorption processes the well-known integro-differential equation (4.1) derived by Ward and Tordai [3] is used. It is the most general relationship between the dynamic adsorption r(t) and the subsurface concentration e(0,t) for fresh non-deformed surfaces and is valid for kinetic-controlled, pure diffusion-controlled and mixed adsorption mechanisms. For a diffusion-controlled adsorption mechanism Eq. (4.1) predicts different F dependencies on t for different types of isotherms. For example, the Frumkin adsorption isotherm predicts a slower initial rate of surface tension decrease than the Langmuir isotherm does. In section 4.2.2. it was shown that reorientation processes in the adsorption layer can mimic adsorption processes faster than expected from diffusion. In this paragraph we will give experimental evidence, that changes in the molar area of adsorbed molecules can cause sueh effectively faster adsorption processes. 

A molecular dynamics approach can also be used to predict mixed gas diffusivities in microporous materials, at the expense of computation cost (e.g., Qureshi and Wei, 1990 Chitra and Yashonath, 1995 Trout et al., 1997 Snurr and Karger, 1997). The empirical correlation of Vignes (1966) for binary diffusivities in liquid solutions and also metallic alloys has been used extensively for calculating binary diffusivities, using the Maxwell-Stefan formalism for flux equations (e.g., Krishna, 1990). 

For the transport of gas mixtures, the generalised Maxwell-Stefan equation (Krishna and WesseUngh, 1997) has been widely adopted to describe multi-component diffusion. Although quantitative descriptions of gas diffusion in various microporous or mesoporous ceramic membranes based on statistical mechanics theory (Oyama et al., 2004) or molecular dynamic simulation (Krishna, 2009) have been reported, the prediction of mixed gas permeation in porous ceramic membranes remains a challenging task, due to the difficulty in generating an accurate description of the porous network of the membrane. 

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