Plug Flow with Dispersion

Dispersion is the enhanced mixing of material through spatial variations in velocity. When it is of interest (when we are not keeping track of the three-dimensional mixing), dispersion is typically one or two orders of magnitude greater than turbulent diffusion. The process of dispersion is associated with a spatial mean velocity. The means used in association with diffusion, turbulent diffusion, and dispersion are identified in Table 6.2. 

Process Variable representing process Mean Scaie of mean 

Turbulent diffusion Tnrbnfent diffnsion coefficient Temporai Minntes 

Dispersion Dispersion coefficient Spatiai Scaie of flow 

A similar spatial mean velocity (bulk mean velocity) is used for the plug flow reactor model. Thus, plug flow with dispersion is a natural match, where the mixing that truly occurs in any reactor or environmental flow is modeled as dispersion. This is the model that will be applied to utilize dispersion as a mixing model. 

---

Plug Flow with Dispersion - Plug flow with dispersion is a concept that is often used to describe one-dimensional flow systems. It is somewhat more flexible in computational models because the mixing within the system is not dependent on reactor size, as with complete mix tanks in series. Plug flow with dispersion will be described in the second half of this chapter because special techniques are needed for the analysis. 

It is interesting to note that Figure 6.6 approaches a Gaussian distribution as more tanks are placed in the series. In fact, equation (6.12) has a dimensionless variance, which will prove useful in comparisons of tanks-in-series and plug flow with dispersion (which will be discussed later in this chapter). 

The one-dimensional mass transport equation for plug flow with dispersion, and a... 

Figure 6.13. Response of the plug flow with dispersion model to a pulse input.

The response of a plug flow with dispersion model to a pulse input, equation (6.42), is given in Figure 6.13 for various values of the Peclet number, Pe = UL/Dl. Equation (6.41) will be applied to the analysis of a pulse with dispersion in a reactor in Example 6.10. 

The assumption that Cou = 1 in equation (6.43) is really only accurate when Pe > 10. The only way to apply this tracer curve to the plug flow with dispersion model while Cou 1 would be to route each portion of the tracer curve through the reactor. With Pe = 9.4, this solution will be close, although stiU an approximation. 

Now, we need a solution to the plug flow with dispersion model for steady-state operation of an air-stripping tower. The mass transport equation for this situation, assuming minimal trichloroethylene builds up in the bubble, is... 

The plug flow with dispersion model results in a degradation to 3.4% of the inflow trichloroethylene concentration. This is significantly different than the plug flow model (1.0%). It is also a more accurate solution. Whether it is the tail of a tracer pulse or a reaction that approaches complete degradation, one needs to be careful about applying the plug flow model when low concentrations, relative to the inflow, are important. 

Both the tanks-in-series and the plug flow with dispersion models have an effective dispersion that we have related to the variance of the tracer cloud ... 

This is shown in Figure 6.14, where the tanks-in-series model with and without end effects is compared with plug flow with dispersion. Above a Peclet number of 10,... 

Rivers are close to the perfect environmental flow for describing the flow as plug flow with dispersion. The flow is confined in the transverse and vertical directions, such that a cross-sectional mean velocity and concentration can be easily defined. In addition, there is less variation in rivers than there is, for example, in estuaries or reactors - both of which are also described by the plug flow with dispersion model. For that reason, the numerous tracer tests that have been made in rivers are useful to characterize longitudinal dispersion coefficient for use in untested river reaches. A sampling of the dispersion coefficients at various river reaches that were... 

P(0) versus 0 for plug flow with dispersion number D,... 

Fig. 10.10 Cell model to represent cross-flow pattern on distillation tray. Ethyiene glycoi synthesis top row), methyl formate synthesis (bottom row). Well-mixed liquid (left) and liquid plug flow with dispersion for four cells (right)... 

A vertical assembly of elements, in conjunction with a high rate of liquid circulation, can represent plug flow with dispersion of the upward flowing gas and the high level of liquid backraixing typical of a bubble column. Plate columns and stirred vessels can be likewise represented by suitable assemblies as shown in Fig. 6. If a generalised sub-routine for the gas-liquid element were available, assemblies like those in Fig. 6 can be readily solved with a standard flowsheeting system. 

The main differential R.T.D. models are schematized in Figure 1. The plug flow model (Figure la) in which each volume element is assumed to have an equal residence time the plug flow with dispersion (P.D.) model (Figure lb) in which the observed residence time dispersion is represented by a single Fick s law-type mechanism (parameter BOj ) superimposed onto the plug flow (3) ... 

---