Residence dispersion model

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

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. 

Solution The first step in the solution is to find a residence time function for the axial dispersion model. Either W t) or f(t) would do. The function has Pe as a parameter. The methods of Section 15.1.2 could then be used to determine which will give the desired relationship between Pe and... 

Example 15.10 Use residence time theory to predict the fraction unreacted for a closed reactor governed by the axial dispersion model. 

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. 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

Determine the dimensionless variance of the residence time distribution in Problem 15.1. Then use Equation (15.40) to fit the axial dispersion model to this system. Is axial dispersion a reasonable model for this situation ... 

Residence time distribution curves for dispersion model. 

The dispersion model is one of the frequently used models. It describes the dispersion of the residence time of the phases according to Fig. 9.16, for example, in one-dimensional flows by superimposing the plug profile of the basic flow with a stochastic dispersion process in axial direction, which is constructed by analogy to Pick s first law of molecular diffusion ... 

Fig. 9.17 Residence times of (a) continuous and (b) drop phase in a liquid pulsed sieve tray extractor according to measurements (squares and crosses) and according to calculations by the dispersion model (drawn lines). Measured and calculated response curves agree well for the continuous phase, but not for the dispersed phase. (From Ref. 14.)...

Plug Flow Reactor with Dispersion. The residence time is still 15 min. The plug flow with a dispersion model gives equation (6.43) ... 

Although it also is subject to the limitations of a single characterizing parameter which is not well correlated, the Peclet number, the dispersion model predicts conversions or residence times unambiguously. For a reaction with rate equation rc = fcC , this model is represented by the differential equation... 

Because actual exposure measurements are often unavailable, exposure models may be used. For example, chemical emission and air dispersion models are used in air quality studies to predict the air concentrations for down-wind residents. Residential wells located down-gradient from a site may not currently show signs of contamination, but they may become contaminated in the future as chemicals in the groundwater migrate to the well site. In these situations, groundwater transport models may estimate the period of time that chemicals of potential concern will take to reach the wells. 

A plot of [log(l/F(s))j 1 versus s[log(l/F(s))]-2 from the above equation should yield a straight line, if the axial dispersion model is applicable. The slope and intercept, yield Jhe values, of the average residence time t and Peclet number Pe. 

FIGURE 7.13 Residence time distribution for various extents of back mixing as predicted by the dispersion model. From Levenspiel [9]. Copyri t 1972 by John Wiley Sons, Inc. Reprinted by permission of John Wiley Sons, Inc. 

Tritium measurements are frequently used to calculate recharge rates, rates or directions of subsurface flow, and residence times. For these purposes, the seasonal, yearly, and spatial variations in the tritium content of precipitation must be accurately assessed. This is difficult to do because of the limited data available, especially before the 1960s. For a careful discussion of how to calculate the input concentration at a specific location, see Michel (1989) and Plummer et al. (1993). Several different approaches (e.g., piston-flow, reservoir, compartment, and advective-dispersive models) to modeling tritium concentrations in groundwater are discussed by Plummer et al. (1993). The narrower topic of using environmental isotopes to determine residence time is discussed briefly below. 

An alternative model for real flows is the dispersion model with the model parameters Bodenstein number (Bo) and mean residence time t, The Bodenstein number which is defined as Bo = uL/D characterises the degree of backmixing during flow. The parameter D is called the axial dispersion coefficient, u is a velocity and L a length. The RTD of the dispersed plug flow model ranges from PFR at one extreme (Bo = °) to PSR at the other (Bo = 0). The transfer function for the dispersion model with closed-closed boundaries is [10] ... 

For the SCC of type II an example of a RTD modelled is shown in Figure 7, The model used is the dispersion model (sec Esq. 6). The values of the model parameters determined arc a Bodenstein number of 8.8 and a mean residence time of 0.6 s. It clearly shows that the model for the RTD explains the frequency response measurement up to a frequency of 2 Hz, At the frequency of 2 Hz the signal-to-noise ratio of 100 is reached. Any mixing processes which affect the transfer function above this frequency cannot be identified. 

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