Of residence time models

The dimensionless variance has been used extensively, perhaps excessively, to characterize mixing. For piston flow, a = 0 and for a CSTR, a = l. Most turbulent flow systems have dimensionless variances that lie between zero and 1, and cr can then be used to fit a variety of residence time models as will be discussed in Section 15.2. The dimensionless variance is generally unsatisfactory for characterizing laminar flows where > 1 is normal in liquid systems. 

Fig. 8. Combined flow reactor models (a) parallel flow reactors with longitudinal diffusion (diffusivities can differ), (b) internal recycle—cross-flow reactor (the recycle can be in either direction), comprising two countercurrent plug-flow reactors with intercormecting distributed flows, (c) plug-flow and weU-mixed reactors in series, and (d) 2ero-interniixing model, in which plug-flow reactors are parallel and a distribution of residence times dupHcates that...

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Ross (R2) measured liquid-phase holdup and residence-time distribution by a tracer-pulse technique. Experiments were carried out for cocurrent flow in model columns of 2- and 4-in. diameter with air and water as fluid media, as well as in pilot-scale and industrial-scale reactors of 2-in. and 6.5-ft diameters used for the catalytic hydrogenation of petroleum fractions. The columns were packed with commercial cylindrical catalyst pellets of -in. diameter and length. The liquid holdup was from 40 to 50% of total bed volume for nominal liquid velocities from 8 to 200 ft/hr in the model reactors, from 26 to 32% of volume for nominal liquid velocities from 6 to 10.5 ft/hr in the pilot unit, and from 20 to 27 % for nominal liquid velocities from 27.9 to 68.6 ft/hr in the industrial unit. In that work, a few sets of results of residence-time distribution experiments are reported in graphical form, as tracer-response curves. 

Fig. 11. Change in concentration with the radius of the residence-time model for con-stant gas holdup and varying contact time and reaction rate [after Gal-Or and Resnick G2,G6)].

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. 

This section describes residence time models that are based on a hydrodynamic description of the process. The theory is simplified but the resulting models still have substantial utility as conceptual tools and for describing some real flow systems. 

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. 

Figure 5.21 Comparison of reaction product concentrations (symbols) and model (solid line) as a function of residence time given in [6J. Toluene-2,4-diisocyanate (TDI) ...

In principle, any type of residence time distribution can be described by combinations of tanks in series or in parallel. This type of modelling of residence time distribution can very easily be implemented in simulation programs by adding the various tanks and adjunct flow streams. 

Except for the case of an ideal plug flow reactor, different fluid elements will take different lengths of time to flow through a chemical reactor. In order to be able to predict the behavior of a given piece of equipment as a chemical reactor, one must be able to determine how long different fluid elements remain in the reactor. One does this by measuring the response of the effluent stream to changes in the concentration of inert species in the feed stream—the so-called stimulus-response technique. In this section we will discuss the analytical form in which the distribution of residence times is cast, derive relationships of this type for various reactor models, and illustrate how experimental data are treated in order to determine the distribution function. 

Since these two types of processes have drastically different effects on the conversion levels achieved in chemical reactions, they provide the basis for the development of mathematical models that can be used to provide approximate limits within which one can expect actual isothermal reactors to perform. In the development of these models we will define a segregated system as one in which the first effect is entirely responsible for the spread in residence times. When the distribution of residence times is established by the second effect, we will refer to the system as mixed. In practice one encounters various combinations of these two limiting effects. 

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. 

Backmix flow (BMF) is the flow model for a CSTR, and is described in Section 2.3.1. BMF implies perfect mixing and, hence, uniform fluid properties throughout the vessel. It also implies a continuous distribution of residence times. The stepwise or discontinuous change in properties across the point of entry, and the continuity of property behavior across the exit are illustrated in Figure 2.3. 

Laminar flow (LF) is also a form of tubular flow, and is the flow model for an LFR. It is described in Section 2.5. LF occurs at low Reynolds numbers, and is characterized by a lack of mixing in both axial and radial directions. As a consequence, fluid properties vary in both directions. There is a distribution of residence times, since the fluid velocity varies as a parabolic function of radial position. 

Vazquez, A. and Calvelo, A., Modelling of residence times in continuous fluidized bed freezers, /. Food Set., 48 (1983b) 1081-1085. 

Dekker et al. [170] have also shown that the steady state experimental data of the extraction and the observed dynamic behavior of the extraction are in good agreement with the model predictions. This model offers the opportunity to predict the effect of changes, both in the process conditions (effect of residence time and mass transfer coefficient) and in the composition of the aqueous and reverse micellar phase (effect of inactivation rate constant and distribution coefficient) on the extraction efficiency. A shorter residence time in the extractors, in combination with an increase in mass transfer rate, will give improvement in the yield of active enzyme in the second aqueous phase and will further reduce the surfactant loss. They have suggested that the use of centrifugal separators or extractors might be valuable in this respect. 

In a real free-radical polymerization, the probability that a given monomer unit in the polymer bears a branch is not constant, but increases with time. Mullikin and Mortimer (tOt) have extended their model to take into account the time-dependence of branching probability and the distribution of residence times in a continuous reactor. They assume that the branching probability b is given by ... 

Figures 6.19(a-d) show the four different types of bifurcation diagram where the stationary-state extent of reaction is plotted as a function of residence time for the model with the uncatalysed reaction included for the special case of no catalyst inflow, 0Q = 0. Three patterns have been seen before (a) unique, (b) isola, and (c) mushroom, in the absence of the un-...

Correct modeling of the flow near the front of a stream requires a rigorous solution of the hydrodynamic problem with rather complicated boundary conditions at the free surface. In computer modeling of the flow, the method of markers or cells can be used 124 however this method leads to considerable complication the model and a great expenditure of computer time. The model corresponds to the experimental data with acceptable accuracy if the front of the streamis assumed to be flat and the velocity distribution corresponds to fountain flow.125,126 The fountain effect greatly influences the distribution of residence times in a channel and consequently the properties of the reactive medium entering the mold. 

This paper presents the first experimental results for the solid motion inside a pilot-scale rotary kiln. Such data are useful to enable efficient pyrolysis reactions inside a rotary kiln to be carried out, through the prediction of residence time and material hold-up, and the evaluation of different surfaces and temperature profiles according to the operating conditions. In the first part, the pilot-scale rotary kiln and the principle of the experiments will be described. An original dynamic solid motion model will be presented in the second part, this dynamic model is derived from the original static model of Seaman [1], The static and dynamic experimental results are finally compared with the simulated results. 

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