Some Fluid Dynamics

Fluid flow, since it transports material, is enmeshed with diffusion in electrochemical cells. Some basics are therefore in order here. 

Dieter Britz Digital Simulation in Electrochemistry, Lect. Notes Phys. 666, 235—246 (2005) www.springerllDk.caD -c- Springer-Verlag Berlin Heidelberg 2005 

The above holds only for laminar flows, which means that the Reynolds number is sufficiently small. It is defined as 

For electrochemical purposes, where electrodes are (usually) embedded in the channel bottom, it is convenient to shift the y coordinate so that y is zero at the channel bottom. The equation for the velocity profile then changes to 

Since the velocity of flow has a parabolic function, the velocity profile near the walls is nonlinear. However, in many works, this is approximated by a linearised form, as the gradient right at the walls. This makes the mathematical analysis of diffusion near one of the walls easier. Differentiating (13.1) and setting y = —h (that is, considering the bottom surface), we obtain 

Strutwolf, Digital Simulation in Electrochemistry, Monographs 

If we ignore entry effects at the inlet end of this channel (see below) and if the flow is laminar (that is, not turbulent), then there is a steady state flow, with no velocity components in the y-direction. The component in the A -direction will then be a known function of y. At the walls (y = h), the fluid clings to the solid surface, that is Vx( h) = 0. As can be shown [4, 5], the velocity profile is of the parabolic form 

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In addition, when the inverse problem is solved, namely the volume-average rates and kernels are identified by comparison between experiments and model predictions under the assumption of perfect mixing (or, in other words, by using Eq. (7.146)), one must remember that the volume-average rates and kernels are not truly kinetic parameters but contain some fluid-dynamic factors in them. This is why, on transferring these rates and kernels to different operating conditions or from one system to another, very poor performances can be observed. 

Therefore, it is very important to predict conversion efficiency, gas temperature distribution, and catalyst temperature distribution for a given condition based on some fundamental data on catalysts. With this in mind, some calculations of catalytic combustion for a gas turbine combustor are carried out based on heat and mass transfer theory with some fluid dynamics aspects. 

On the basis of some empiricism and some fluid dynamics, Kunii and Levenspiel proposed the following formulas for estimating these quantities ... 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

In writing the Lagrangean density of quantum mechanics in the modulus-phase representation, Eq. (140), one notices a striking similarity between this Lagrangean density and that of potential fluid dynamics (fluid dynamics without vorticity) as represented in the work of Seliger and Whitham [325]. We recall briefly some parts of their work that are relevant, and then discuss the connections with quantum mechanics. The connection between fluid dynamics and quantum mechanics of an electron was already discussed by Madelung [326] and in Holland s book [324]. However, the discussion by Madelung refers to the equations only and does not address the variational formalism which we discuss here. 

In theory it should be possible to calculate the capture efficiency without measurements. Some attempts have used computational fluid dynamics (CFD) models, but difficulty modeling air movement and source characteristics have shown that it will be a long time before it will be possible to calculate the capture efficiency in advance. ... 

We begin our discussion of LG computers by considering some of the generic advantages and disadvantages of using LGs to simulate fluid dynamics. Later on ill this section we will provide a brief overview of some of the more popular LG computers that are now in use. 

Additionally, a personal objective was to provide the information contained within this book in such a way that it could be used regularly in the field rather than be relegated to a bookshelf with other works of occasional reference. As such, although this book is essentially concerned with applied chemistry, I found it necessary to devote several of the initial chapters to a discussion on some basic but practical engineering aspects. Subjects covered include fluid dynamics, thermodynamics, the various types and designs of boilers to be found, and the function of all the critical system auxiliaries and components. The subject of boiler water chemistry is so inextricably bound up with the mechanical operation of boiler plants and all their various systems and subsystems that it is impossible to discuss one topic without the other. 

The value of the coefficient will depend on the mechanism by which heat is transferred, on the fluid dynamics of both the heated and the cooled fluids, on the properties of the materials through which the heat must pass, and on the geometry of the fluid paths. In solids, heat is normally transferred by conduction some materials such as metals have a high thermal conductivity, whilst others such as ceramics have a low conductivity. Transparent solids like glass also transmit radiant energy particularly in the visible part of the spectrum. 

While some of these dynamics are natural, others are due to technological systems, or to cultural/social behavior or changes. While some of them can be described with high precision (e.g., fluid dynamics in a pipe or WWTP efficiencies), others can be approached (e.g., prevision of water demand) or estimated with low precision (e.g., rainfall regime of next years). 

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

In practice, the process regime will often be less transparent than suggested by Table 1.4. As an example, a process may neither be diffusion nor reaction-rate limited, rather some intermediate regime may prevail. In addition, solid heat transfer, entrance flow or axial dispersion effects, which were neglected in the present study, may be superposed. In the analysis presented here only the leading-order effects were taken into account. As a result, the dependence of the characteristic quantities listed in Table 1.5 on the channel diameter will be more complex. For a detailed study of such more complex scenarios, computational fluid dynamics, to be discussed in Section 2.3, offers powerful tools and methods. However, the present analysis serves the purpose to differentiate the potential inherent in decreasing the characteristic dimensions of process equipment and to identify some cornerstones to be considered when attempting process intensification via size reduction. 

In computational fluid dynamics only the last two methods have been extensively implemented into commercial flow solvers. Especially for CFD problems the FVM has proven robust and stable, and as a conservative discretization scheme it has some built-in mechanism of error avoidance. For this reason, many of the leading commercially available CFD tools, such as CFX4/5, Fluent and Star-CD, are based on the FVM. The oufline on CFD given in this book wiU be based on this method however, certain parts of the discussion also apply to the other two methods. 

Computational fluid dynamics (CFD) programs are more specialized, and most have been designed to solve sets of equations that are appropriate to specific industries. They can then include approximations and correlations for some features that would be difficult to solve for directly. Four major packages widely used are Fluent (http //www.fluent.com/), CFX (now part of ANSYS), Comsol Multiphysics (formerly FEMLAB) (http //www.comsol.com/), and ANSYS (http //www.ansys.com/). Of these, Comsol Multiphysics is particularly useful because it has a convenient graphical-user interface, permits easy mesh generation and refinement (including adaptive mesh refinement), allows the user to add phenomena and equations easily, permits solution by continuation methods (thus enhancing... 

If one wants to model a process unit that has significant flow variation, and possibly some concentration distributions as well, one can consider using computational fluid dynamics (CFD) to do so. These calculations are very time-consuming, though, so that they are often left until the mechanical design of the unit. The exception would occur when the flow variation and concentration distribution had a significant effect on the output of the unit so that mass and energy balances couldn t be made without it. 

Hydrotreating units in the past were built with one reactor, to meet the 500 ppm S specification however, a second reactor is needed to cope with the actual requirements. Several attempts have been made also to change the operating conditions to improve performance of the units and to achieve the stipulated level of desulfurization. From a co-current fluid-dynamics, the first modification went into countercurrent feeding. In a countercurrent reactor, where hydrogen is fed at one end and the feed in the other, the most difficult to-desulfurize compounds, will react under the higher hydrogen concentration. The countercurrent operation introduces some other problems, such as hot spots and vapor-liquid contact. 

Another Lagrangian-based description of micromixing is provided by multienvironment models. In these models, the well macromixed reactor is broken up into sub-grid-scale environments with uniform concentrations. A four-environment model is shown in Fig. 5.16. In this model, environment 1 contains unmixed fluid from feed stream 1 environments 2 and 3 contain partially mixed fluid and environment 4 contains unmixed fluid from feed stream 2. The user must specify the relative volume of each environment (possibly as a function of age), and the exchange rates between environments. While some qualitative arguments have been put forward to fit these parameters based on fluid dynamics and/or flow visualization, one has little confidence in the general applicability of these rules when applied to scale up or scale down, or to complex reactor geometries. 

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. 

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