Chemical residence-time distribution

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

An industrial chemical reacdor is a complex device in which heat transfer, mass transfer, diffusion, and friction may occur along with chemical reaction, and it must be safe and controllable. In large vessels, questions of mixing of reactants, flow distribution, residence time distribution, and efficient utilization of the surface of porous catalysts also arise. A particular process can be dominated by one of these factors or by several of them for example, a reactor may on occasion be predominantly a heat exchanger or a mass-transfer device. A successful commercial unit is an economic balance of all these factors. 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

To run the residence time distribution experiments under conditions which would simulate the conditions occurring during chemical reaction, solutions of 15 weight percent and 30 percent polystyrene in benzene as well as pure benzene were used as the fluid medium. The polystyrene used in the RTD experiment was prepared in a batch reactor and had a number average degree of polymerization of 320 and a polydispersity index, DI, of 1.17. 

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. 

Washout experiments can be used to measure the residence time distribution in continuous-flow systems. A good step change must be made at the reactor inlet. The concentration of tracer molecules leaving the system must be accurately measured at the outlet. If the tracer has a background concentration, it is subtracted from the experimental measurements. The flow properties of the tracer molecules must be similar to those of the reactant molecules. It is usually possible to meet these requirements in practice. The major theoretical requirement is that the inlet and outlet streams have unidirectional flows so that molecules that once enter the system stay in until they exit, never to return. Systems with unidirectional inlet and outlet streams are closed in the sense of the axial dispersion model i.e., Di = D ut = 0- See Sections 9.3.1 and 15.2.2. Most systems of chemical engineering importance are closed to a reasonable approximation. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

Recycling to monomers, fuel oils or other valuable chemicals from the waste polymers has been attractive and sometimes the system has been commercially operated [1-4]. It has been understood that, in the thermal decomposition of polymers, the residence time distribution (RTD) of the vapor phase in the reactor has been one of the major factors in determining the products distribution and yield, since the products are usually generated as a vapor phase at a high temperature. The RTD of the vapor phase becomes more important in fluidized bed reactors where the residence time of the vapor phase is usually very short. The residence time of the vapor or gas phase has been controlled by generating a swirling flow motion in the reactor [5-8]. 

Resident Time Distribution (RTD) is widely employed in the chemical engineering industry, as an analytical tool for characterizing flow dynamics within reactor vessels. RTD provides a quantitative measure of the back-mixing with in a reactor system [2]. However the cost and time involved in building and operating a pilot- or full scale reactor for RTD analysis can be economically prohibitive. As such we have implemented a numerical RTD technique through the FLUENT (ver. 6.1) commercial CFD package. 

Gavrilescu, M. and R.Z. Tudose, Residence time distribution of the liquid phase in a concentric-tube airlift reactor. Chemical Engineering and Processing, 1999. 38(3) p. 225-238. 

Chemical Kinetics, Tank and Tubular Reactor Fundamentals, Residence Time Distributions, Multiphase Reaction Systems, Basic Reactor Types, Batch Reactor Dynamics, Semi-batch Reactors, Control and Stability of Nonisotheimal Reactors. Complex Reactions with Feeding Strategies, Liquid Phase Tubular Reactors, Gas Phase Tubular Reactors, Axial Dispersion, Unsteady State Tubular Reactor Models... 

The available models mostly refer to ideal reactors, STR, CSTR, continuous PFR. The extension of these models to real reactors should take into account the hydrodynamics of the vessel, expressed in terms of residence time distribution and mixing state. The deviation of the real behavior from the ideal reactors may strongly affect the performance of the process. Liquid bypass - which is likely to occur in fluidized beds or unevenly packed beds - and reactor dead zones - due to local clogging or non-uniform liquid distribution - may be responsible for the drastic reduction of the expected conversion. The reader may refer to chemical reactor engineering textbooks [51, 57] for additional details. 

In general, each form of ideal flow can be characterized exactly mathematically, as can the consequences of its occurrence in a chemical reactor (some of these are explored in Chapter 2). This is in contrast to nonideal flow, a feature which presents one of the major difficulties in assessing the design and performance of actual reactors, particularly in scale-up from small experimental reactors. This assessment, however, may be helped by statistical approaches, such as provided by residence-time distributions. It... 

Several age-distribution functions may be used (Danckwerts, 1953), but they are all interrelated. Some are residence-time distributions and some are not. In the discussion to follow in this section and in Section 13.4, we assume steady-flow of a Newtonian, single-phase fluid of constant density through a vessel without chemical reaction. Ultimately, we are interested in the effect of a spread of residence times on the performance of a chemical reactor, but we concentrate on the characterization of flow here. 

Consider the steady flow of fluid at a volumetric rate q through a stirred tank as a closed vessel, containing a volume V of fluid, as illustrated in Figure 13.4. We assume the flow is ideal in the form of BMF at constant density, and that no chemical reaction occurs. We wish to derive an expression for E(t) describing the residence-time distribution (RTD) for this situation. 

Rough correlations of Peclet numbers for dispersion are given by Wen (in Petho Noble, Residence Time Distribution in Chemical Engineering, 1982)... 

Evaluate the conversion for first-order reaction from a tracer pulse response curve using the method in example CSTRPULSE. Show that although the residence time distributions may be the same in the two cases, the overall chemical conversion is not, excepting for the case of first-order reaction. 

For isothermal, first-order chemical reactions, the mole balances form a system of linear equations. A non-ideal reactor can then be modeled as a collection of Lagrangian fluid elements moving independe n tly through the system. When parameterized by the amount of time it has spent in the system (i.e., its residence time), each fluid element behaves as abatch reactor. The species concentrations for such a system can be completely characterized by the inlet concentrations, the chemical rate constants, and the residence time distribution (RTD) of the reactor. The latter can be found from simple tracer experiments carried out under identical flow conditions. A brief overview of RTD theory is given below. 

The IEM model is a simple example of an age-based model. Other more complicated models that use the residence time distribution have also been developed by chemical-reaction engineers. For example, two models based on the mixing of fluid particles with different ages are shown in Fig. 5.15. Nevertheless, because it is impossible to map the age of a fluid particle onto a physical location in a general flow, age-based models cannot be used to predict the spatial distribution of the concentration fields inside a chemical reactor. Model validation is thus performed by comparing the predicted outlet concentrations with experimental data. 

Most chemical reaction engineering textbooks contain material on residence time distribution theory. Levenspiel [17] and Hill [18] present particularly useful introductions as do refs. 9 and 16. The proceedings of a recent summer school [19] contains a brief overview of the field [8] as well as papers describing many specific applications of RTD theory in chemical engineering contexts. Nauman s comprehensive invited review cited earlier [4] is an extremely thorough and yet highly readable contribution to the literature. The book by Nauman and Buffham [20], will no doubt fill a most important gap in the literature on mixing in continuous flow systems. 

The purpose of tracer experiments is to extract information about the system in a chemical reaction engineering context, it is the mixing within the system which is of interest, as represented by the system residence time distribution. Because flow mixing is an inherently linear process, the exact form of the RTD which is recovered from a tracer response experiment should be independent both of the amount of tracer used in the test and also of the particular functional form in which the tracer was... 

These relationships are of profound importance for, once a reactor has been described by means of a transfer function, they enable the residence time distribution for that reactor to be chsiracterised in terms of its mean, variance, skewness, etc. Such a characterisation in terms of a few low-order moments is often entirely adequate for the requirements of chemical reaction engineering. 

R. Aris, in A. Petho and R.D. Noble (Eds.), Residence Time Distribution Theory in Chemical Engineering, Verlag Chemie, Weinheim, 1982. 

The notions of different combinations of ideal reactors and residence time distributions are essential in analyzing these problems and in suggesting appropriate solutions. We summarize the many applications of chemical reaction engineering in Figure 8-18, which indicates the types of molecules, reactors, and reactors we can handle. 

The study of nonideal flow and liquid holdup can be done by residence time distribution (RTD) experiments (tracing techniques) or by use of correlations derived from literature. Dining this step, physical mechanisms that are sensitive to size are investigated separately from chemical (kinetic or equilibrium) studies (Trambouze, 1990). Here, the fixed bed is... 

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