Extensions of Residence Time Theory

Stirred tank case, it has been difflcnlt to find experimental evidence of segregation for single-phase reactions. Real CSTRs approximate perfect mixing when observed on the time and distance scales appropriate to indnstrial reactions provided that the feed is premixed. Even with unmixed feed, the experimental observation of segregation requires very fast reactions. The standard assumption of perfect mixing in a CSTR is usually justified. Worry when a highly reactive component is separately fed and when the reaction is sensitive to mixing time. See Section 4.6. 

Following a rapid initial dispersion to the Kolmogorov scale, the packets continue to evolve in size and shape but at a relatively slow rate. Molecular-level mixing occurs by diffusion between packets, and the rates of diffusion and of the consequent chemical reaction can be calculated. Early versions of the model assumed spherical packets of constant and uniform size. Variants now exist that allow the packet size and shape to evolve with time. Regardless of the details, these packets are so small that they typically equilibrate with their environment in much less than a second. This is so fast compared to the usual reaction half-lives and to the mean residence time in the reactor that the vessel behaves as if it were perfectly mixed. 

In laminar flow stirred tanks, the packet diffusion model is replaced by a slab diffusion model. The diffusion and reaction calculations are similar to those for the turbulent flow case. Again, the conclusion is that perfect mixing is almost always a good approximation for all but the fastest reactions. 

The results in this chapter are restricted in large part to steady-state, homogeneous, isothermal systems. More general theories have been developed. The next few sections briefly outline some extensions of residence time theory. 

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Interpretation of tracer data by means of residence time theory, in the extremes of complete and minimum segregation, has been reviewed and extended to treat transient response under reacting conditions. While residence time theory was initially developed for industrial application to nonideal steady state reactors, its transient extension seems especially well suited for describing segments of natural flows which are too complex to interpret using simpler models, such as dispersion. 

Residence Time Theory. Residence time theory is based on the consideration of whether a particle will reach the cyclone wall in a given residence time. In the development of this theory, the distribution of all particles across the inlet is assumed to be homogeneous (Rietema, 1961). The cut size will be the size of those particles that enter the center of the inlet pipe and just reach the wall within the residence time. Using this theory coupled with extensive experimental test data, Rietema was able to estabhsh a set of empirical correlations and suggest a criterion (a characteristic cyclone number) for optimum design of hydrocyclones. 

The concept of residence time distribution (RTD) and its importance in flow processes first developed by Danckwerts (1953) was a seminal contribution to the emergence of chemical engineering science. An introduction to RTD theory is now included in standard texts on chemical reaction engineering. There is also an extensive literature on the measmement, theory, and application of residence time distributions. A literature search returns nearly 5000 references containing the concept of residence time distribution and some 30 000 references dealing with residence time in general. This chapter necessarily provides only a brief introduction the references provide more comprehensive treatments. 

The "condensation theory" presented by Lahaye et al. can account for part of the experimental facts. The model presented below is an extension of that theory. It also includes assumptions on how hydrogen partial pressure and gas phase reactor residence time affect the rate of deposition. 

In this chapter, a number of simplified examples have been outlined to demonstrate how AR theory may be used to answer common reactor synthesis problems related to adiabatic systems and minimum residence time. A number of natural extensions to these discussions may be carried out that enhance the use of AR theory to nonisothermal systems. For the interested reader, two notable papers are available that extend on the ideas discussed here. Nicol et al. (1997) show how the AR for an exothermic reaction may be generated that incorporates external heating or cooling, whereas Glasser et al. (1992) extend the two-dimensional preheating examples, shown in this chapter, to involving x-T-r space. 

Commercial codes, e.g. PowerFLOW, which use lattice-based approaches are available, and this particular code was used in the present work. Based on discrete forms of the kinetic theory equations, this code employs an approach that is an extension of lattice gas and lattice Boltzmann methods in which particles exist at discrete locations in space, and are allowed to move in given directions at particular speeds over discrete time intervals. The particles reside on a cubic lattice composed of voxels, and move from one voxel to another at each time step. Solid surfaces are accommodated through the use of surface elements, and arbitrary surface shapes can be represented. Particle advection, and particle-particle and particle-surface interactions, are all considered at a microscopic level to simulate fluid behaviour in a way which ensures conservation of mass, momentum and energy, and which recovers solutions of the continuum flow... 

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