Complex flow patterns equations

For any more complex flow pattern we must solve the fluid mechanics to describe the fluid flow in each phase, along with the mass balances. The cases where we can still attempt to find descriptions are the nonideal reactor models considered previously in Chapter 8, where laminar flow, a series of CSTRs, a recycle TR, and dispersion in a TR allow us to modify the ideal mass-balance equations. 

However, there exists a way to employ the rigorous equations of continuum mechanics even for the cases, in which real phase boundaries cannot be exactly localized. This way is associated with the idea of hydrodynamic analogy between complex and simpler flow phenomena. More precisely, some particular similarities are meant between complex flow patterns encountered in industrial separations and geometrically simpler flows like planar films, cylindrical jets, spherical drops, etc., as well as their combinations (Kenig, 1997). These similarities are used in the hydrodynamic analogy approach by which the complex hydrodynamics established in a real column is replaced with an appropriate combination of simpler flow patterns. Such a replacement occurs on the basis of experimental observations which are very important for the successful... 

For random mat non-woven reinforcements, permeability is isotropic in-plane while, for other textile structures, the permeability will be different in different directions, depending upon the nature of the textile structure (number and size of tows, warp and weft densities etc.). This differential permeability will result in complex flow patterns in the tool, making flow prediction even more important, although the use of D Arcy s equation then becomes an over-simplification. 

The inside transfer coefficient may be estimated from equation 7.5.7. The shell side coefficient is difficult to estimate because of the complex flow patterns within a baffled shell. For further details see Coulson and Richardson... 

The solution of (7.3.3-1)-(7.3.3-2) is not straightforward. A detailed treatment of the subject is given in Chapter 12 on complex flow patterns. In many cases, a given type of velocity field can be imposed and the corresponding pressure field is calculated from a specific pressure drop equation. 

The flow becomes highly complex in a spiral-wound module containing a feed-side spacer screen. Numerical solutions of the governing equations incorporating most of these complexities have been/are being implemented (Wiley and Fletcher, 2003) using computational fluid dynamics models (see Schwinge et al. (2003) for the complex flow patterns in a spacer-filled channel). 

A classification of dispersion models for fixed-bed tubular reactors, for example, is given by Froment and Hofmann [2]. For more complex flow patterns more elaborated and complete models are required where the flow fields are described via the solution of the Navier-Stokes equations. The understanding of the complex flow phenomena involved as well as the solution of these vector equations make the problem much more difficult to analyze spending reasonable costs and efforts. The advanced reactor models are discussed in the subsequent chapters, only a brief introduction to the idealized reactor models are presented in this chapter as these models are principal tools for chemical reaction engineers. In particular, the idealized models are easy to calculate, and they give the extreme values of the conversions between which those realized in a real reactor will occur provided there is no bypassing of reactants in the reactor. 

It is shown in Section 9.9.5 that, with the existence of various bypass and leakage streams in practical heat exchangers, the flow patterns of the shell-side fluid, as shown in Figure 9.79, are complex in the extreme and far removed from the idealised cross-flow situation discussed in Section 9.4.4. One simple way of using the equations for cross-flow presented in Section 9.4.4, however, is to multiply the shell-side coefficient obtained from these equations by the factor 0.6 in order to obtain at least an estimate of the shell-side coefficient in a practical situation. The pioneering work of Kern(28) and DoNOHUE(lll who used correlations based on the total stream flow and empirical methods to allow for the performance of real exchangers compared with that for cross-flow over ideal tube banks, went much further and. 

These models are designed to define the complex entrance effects and convection phenomena that occur in a reactor and solve the complete equations of heat, mass balance, and momentum. They can be used to optimize the design parameters of a CVD reactor such as susceptor geometry, tilt angle, flow rates, and others. To obtain a complete and thorough analysis, these models should be complemented with experimental observations, such as the flow patterns mentioned above and in situ diagnostic, such as laser Raman spectroscopy. 

The complex three-dimensional flow pattern within the cyclone is dominated by the radial (Fr) and tangential (V0) velocity components. The vertical component is also significant but plays only an indirect role in the separation. The tangential velocity in the vortex varies with the distance from the axis in a complex manner, which can be described by the equation... 

At some point in most processes, a detailed model of performance is needed to evaluate the effects of changing feedstocks, added capacity needs, changing costs of materials and operations, etc. For this, we need to solve the complete equations with detailed chemistry and reactor flow patterns. This is a problem of solving the R simultaneous equations for S chemical species, as we have discussed. However, the real process is seldom isothermal, and the flow pattern involves partial mixing. Therefore, in formulating a complete simulation, we need to add many additional complexities to the ideas developed thus far. We will consider each of these complexities in successive chapters temperature variations in Chapters 5 and 6, catalytic processes in Chapter 7, and nonideal flow patterns in Chapter 8. In Chapter 8 we will return to the issue of detailed modeling of chemical reactors, which include all these effects. 

Thus we see that environmental modeling involves solving transient mass-balance equations with appropriate flow patterns and kinetics to predict the concentrations of various species versus time for specific emission patterns. The reaction chemistry and flow patterns of these systems are sufficiently complex that we must use approximate methods and use several models to try to bound the possible range of observed responses. For example, the chemical reactions consist of many homogeneous and catalytic reactions, photoassisted reactions, and adsorption and desorption on surfaces of hquids and sohds. Is global warming real [Minnesotans hope so.] How much of smog and ozone depletion are manmade [There is considerable debate on this issue.]... 

Mean temperature differences in such flow patterns are obtained by solving the differential equation. Analytical solutions have been found for the simpler cases, and numerical ones for many important complex patterns, whose results sometimes are available in generalized graphical form. 

In some practical kinds of dryers, the flow patterns of gas and solid are so complex that the kind of rate equation discussed in this... 

Chemical engineers were not the pioneers in this field because chemical engineering flow problems can be very complex. Some of the first users of CFD were car, plane and boat designers. One of the reasons for this was that CFD could tell the designers exactly what they wanted to know, that is the flow patterns obtained while their new designs moved. Indeed, the possibility to use Euler s equations for flow description has been one of the major contributions to the development of these applications. These kinds of CFD techniques have also been projected and have been successfully used to analyze heat flow from a body immersed into the flowing fluid [3.29, 3.30]. 

In some practical kinds of dryers, the flow patterns of gas and solid are so complex that the kind of rate equation discussed in this section cannot be applied readily. The sizing of such equipment is essentially a scale-up of pilot plant tests in similar equipment. Some manufacturers make such test equipment available. The tests may establish the residence time and the terminal conditions of the gas and solid. 

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