Residence time theory

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of residence time distributions are important aspects of chemical reaction engineering. Part of the inspiration for residence time theory came from the black box analysis techniques used by electrical engineers to study circuits. These are stimulus-response or input-output methods where a system is disturbed and its response to the disturbance is measured. The measured response, when properly interpreted, is used to predict the response of the system to other inputs. For residence time measurements, an inert tracer is injected at the inlet to the reactor, and the tracer concentration is measured at the outlet. The injection is carried out in a standardized way to allow easy interpretation of the results, which can then be used to make predictions. Predictions include the dynamic response of the system to arbitrary tracer inputs. More important, however, are the predictions of the steady-state yield of reactions in continuous-flow systems. All this can be done without opening the black box. 

Transient experiments with inert tracers are used to determine residence time distributions. In real systems, they will be actual experiments. In theoretical studies, the experiments are mathematical and are applied to a d5mamic model of the system. 

Negative Step Changes and the Washout Function. Suppose that an inert tracer has been fed to a CSTR for an extended period of time, giving C, = Cout = Co for r 0. At time r = 0, the tracer supply is suddenly stopped so that = 0 for r 0. Equation (14.2) governs the transient response of the system. For t 0, 

Tracer molecules originally in the system at time t = 0 gradually wash out. The exponential form of Equation (15.1) is specific to a CSTR, but the concept of washout applies to any flow system. Consider some time t 0 when the fraction of molecules remaining in the system is W(t) = C ut(0ICo- These molecules must necessarily have entered the reactor before time t = 0 since no tracer was fed 

W t) = Fraction of molecules leaving the system that experienced a residence time greater than t 

It is apparent that W(0) = 1 since all molecules must have a residence time of zero or longer and that tV(oo) = 0 since all molecules will eventually leave the system. Also, the function W(t) will be nonincreasing. 

Suppose now that a pilot plant or full-scale reactor has been built and operated. How can its performance be used to confirm the kinetic and transport models and to improve future designs Reactor analysis begins with an operating reactor and seeks to understand several interrelated aspects of actual performance kinetics, flow patterns, mixing, mass transfer, and heat transfer. The present chapter is concerned with the analysis of flow and mixing processes and their interactions with kinetics. It uses residence time theory as the major tool for the analysis. 

The time that a molecule spends in a reactive system will affect its probability of reacting and the measurement, interpretation, and modeling of RTDs are important 

Chemical Reactor Design, Optimization, and Scaleup, Second Edition. By E. B. Nauman Copyright 2008 John Wiley Sons, Inc. 

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Example 8.1 derived a specific example of a powerful result of residence time theory. The residence time associated with a streamline is t = LIVz. The outlet concentration for this streamline is ahatchit)- This is a general result applicable to diffusion-free laminar flow. Example 8.1 treated the case of a... 

Example 15.9 Use residence time theory to predict the fraction unreacted for an isothermal, homogeneous, first-order reaction occurring in a CSTR and aPFR. 

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

This equation is a result of the residence time theory of particle collection. In this theory, the time that it takes for a particle to reach the wall is balanced by the time that a particle spends in the cyclone. The particle size that makes it to the wall by the time that it exits the cyclone is the particle size collected at 50 percent collection efficiency, Dpth. 

Weiss M. The relevance of residence time theory to pharmacokinetics. Eur J Clin Pharmacol 1992 43 571-9. 

Since the foundations of residence time theory are rigorously given elsewhere (5,6, 7), only those features which are essential to the present treatment will be given here. The residence time distribution (residence time frequency function exit age distribution), f(t), is defined such that f(t)dt is the fraction of fluid at any instant leaving the system, having spent time between t and t + dt within the system. The cumulative residence time distribution is... 

Theoretical considerations of residence time theory imply the following important trends ... 

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 539 The delta function has another integral of substantial use ... 

Fig. 8. Normalized residence time curves for ions of different mass accelerated to a fixed ion exit energy of 6.8 eV under conditions of a dc repeller field. Plotted is the relative ion intensity having a relative residence time greater than r/t, where t is the average residence time. Theory predicts that the shape of the curve is not mass-dependent, and experiment confirms this. The theoretical curve is computed for a Gaussian electron-beam distribution, with a full-width at half-maximum equal to the dimension of the slit through which the electron beam enters the source. Corrections resulting from the initial Maxwellian velocity distribution of the ions are ignored since they are negligible.

Nauman, E.B. (2008). Residence time theory. Industrial Engineering Chemistry... 

Each theory in this category offers a relatively simple correlation for the static pressure drop and the cut size of a hydrocyclone described by a few (but often not all) dimensions. The theories fall into two main groups the equilibrium orbit theory and the residence time theory. 

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