Plug Flow PF

Consider an element of fluid (as tracer ) entering the vessel at t = 0. Visualizing what happens to the element of fluid is relatively simple, but describing it quantitatively as Eft) requires an unusual mathematical expression. The element of fluid moves through the vessel without mixing with fluid ahead of or behind it, and leaves the vessel all at once at a time equal to the mean residence time ft = Vlq for constant density). Thus, Eft) = 0 for 0 t f, but what is Eft) at t = F  

Since () / , the area under the vertical line representation for PF is unity, as required by equation 13.3-1. 

Another important property of the delta function is, for any function g(t)  

We use the property of the delta function contained in equation 13.4-10, and the definitions of 6 and crjj in the 6 analogues of equations 13.3-14 and 13.3-16a, respectively, to obtain 

The second result confirms our intuitive realization that there is no spread in residence time for PF in a vessel. 

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Ideal flow is introduced in Chapter 2 in connection with the investigation of kinetics in certain types of ideal reactor models, and in Chapter 11 in connection with chemical reactors as a contrast to nonideal flow. As its name implies, ideal flow is a model of flow which, in one of its various forms, may be closely approached, but is not actually achieved. In Chapter 2, three forms are described backmix flow (BMF), plug flow (PF), and laminar flow (LF). 

In writing Equation 16.17, plug flow (PF) was assumed for both the gas and liquid phases. Thus this equation should be valid for packed columns. 

A particularly important consideration in designing contactors for gas-liquid reactions is the validity of the ideal flow assumption [plug flow (PF) or mixed flow (MF)] normally used in the design. Numerous studies have been reported on the role of nonideality (i.e., backmixing) in gas-liquid contactors, and based on these, some qualitative guidelines... 

Model discrimination was done comparing simulated and 41 experimental data sets. The measure of model accuracy was defined as the sum of squares of relative errors of prediction for Sc2+ and XcH4- The bubble assemblage model assuming plug flow (PF) in the horizontally distributor zone (BAM+PF) was best (for details see [49]). A comparison between results of simulations obtained with the BAM + PF model and experimental data is shown in Figures 25 and 26. C2+ selectivity is very well predicted the conversion of methane is also predicted quite well, however, a tendency to underestimate the conversion was observed. The standard deviations for the prediction of Sc2+ and Xch4 amount to 7 and 8 % respectively. It should be emphasized that the simulations were done without any tuned parameters. 

In this section, we develop two simple models, each of which has one adjustable parameter the tanks-in-series (TIS) model and the axial-dispersion or dispersed-plug-flow (DPF) model. We focus on the description of flow in terms of RTD functions and related quantities. In principle, each of the two models is capable of representing flow in a single vessel between the two extremes of BMF and PF. 

In PF, the transport of material through a vessel is by convective or bulk flow. All elements of fluid, at a particular axial position in the direction of flow, have the same concentration and axial velocity (no radial variation). We can imagine this ideal flow being blurred or dispersed by backmixing of material as a result of local disturbances (eddies, vortices, etc.). This can be treated as a diffusive flow superimposed on the convective flow. If the disturbances are essentially axial in direction and not radial, we refer to this as axial dispersion, and the flow as dispersed plug flow (DPF). (Radial dispersion may also be significant, but we consider only axial dispersion here.)... 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

A number of driers may be used in series. For example, a cascade of well-mixed driers approximafes fo fhe performance of a plug flow drier. However, of more pracfical significance is the combination of a well-mixed drier followed by a plug flow drier for drying very wet solids which may defluidize if fed directly to a PF unit. The CSTR drier is better able to cope with the high moisture content feed and the plug flow drier allows a low final moisture content to be obtained. 

The models of flow dispersion are based on the plug flow model. However, in comparison with the PF model, the dispersion flow model considers various perturbation modes of the piston distribution in the flow velocity. If the forward and backward perturbations present random components with respect to the global flow direction, then we have the case of an axial dispersion flow (ADF). In addition, the axial and radial dispersion flow is introduced when the axial flow perturbations are coupled with other perturbations that induce the random fluid movement in the normal direction with respect to the global flow. 

Note PF = plug flow PM = perfect mixing (continuous stirred lank)... 

The physical arrangement involves a method of containing the solid reactant while the fluid phase is passed over the solid in such a way that intimate contact is possible between the flowing fluid and the resident solid. This physical arrangement is familiar from the static bed plug flow catalytic reactor and is now applied here to a reactor where solid interactions with a fluid phase, including adsorption, are studied. The interactions between the phases involved in a catalytic TS-PFR have been described above. The non-catalytic interactions are now considered as they take place in a TS-PF-SSR. The differences between the two reactor types lie in hardware requirements and in the mode of operation. 

There are two kinds of TS-SSR the plug flow version which we have called the TS-PF-SSR, and the back-mix version which we call the TS-CST-SSR. The modes of operation, theoretical underpinnings and data processing methods appropriate for each of these two SSR types are described in Chapter 5. The hardware is described below. 

PF-FBD Plug flow FBD M-FBD Multistage FBD IT-FBD Immersed tubes FBD IS-FBD Inert solids FBD... 

Figure 6.36. Plot of the dimensionless concentration of cell mass x and substrate s for a continuous culture as a function of the dimensionless mean residence time I as in Fig. 6.1b with Xq > 0 Calculated comparison between a CSTR with maximum mixing (ST m) or one with total segregation (ST J and a continuous plug flow reactor (PF), assuming Monod kinetics with a death rate (Tsai et al., 1969).

AD, axial dispersion MC, mixing cells PF, plug flow PM, perfectly mixed. 

We now calculate the difference between this inlet concentration and the outlet concentration Cout (t). for two quite different situations plug flow in our system (PF), and perfect mixing (CSTR for continuous stirred tank reactor). In both cases the mean residence time is Tm = V"/Q. The results are ... 

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