Dispersion small deviation from plug flow

Figure4.10 Ffunctions (step response curves) for different values of Bo = uL/D for a small extent of dispersion (small deviation from plug flow). Adapted from Levenspiel (1999).

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

In the preceding section we discussed the dispersion model which can account for small deviations from plug flow. It happens that a series of perfectly mixed tanks (backmix flow) will give tracer response curves that are somewhat similar in shape to those found from the dispersion model. Thus, either type of model could be used to correlate experimental tracer data. 

In conclusion then, it would seem that for small deviations from plug flow, both the dispersion and the tanks-in-series models will give satisfactory results. Up to the present, which one is used may be largely a matter of personal preference. 

Dl/uL -> 0), and so for many practical cases of interest a comparison is possible. Thus, for small deviations from plug flow, either the dispersion model or stirred tanks model may be satisfactorily used depending on one s personal preference. 

Dispersion In tubes, and particularly in packed beds, the flow pattern is disturbed by eddies Amose effect is taken into account by a dispersion coefficient in Pick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of . Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reactor with a small deviation from plug flow, without specifying the magnitude of small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The RTD in the tubular reactor was determined experimentally with water as fluid and Brilliant Blue dye as tracer. The tracer was introduced at the reactor inlet in the form of a step function. The concentration of the dye was measured with an UV-vis spectrometer and the response curve is given as f-curve. As the experimental f-curve shown in Figure 3.29 is very steep, a low axial dispersion can be expected. Therefore, RTD will be described with the dispersion model supposing small deviation from plug flow (Equation 3.50). The f-curve valid for small dispersion Bo > 100) can be obtained by integrating the RTD given by (0) (Equation 3.50). 

In the case of small deviations from plug flow or simple dispersion within the reactor volume, one-parameter models may suffice to account for the RTD. Two of them are widely used ... 

This simply assumes that axial dispersion (D m. s ) is superimposed onto plug flow. Axial dispersion may be caused by a velocity profile in the radial direction or statistical dispersion in a packing or turbulent diffusion or by any physicochemical process which delayes some particles with respect to others. The model parameter is the axial PECLET number, Pe = uL/D, or its reciprocal, the dispersion number, D /uL. Depending on the boundary conditions assumed at the reactor inlet and outlet (which are different from those of the simple assumptions above), a lot of mathematical formulae can be found in the literature for the RTD [3]. This is often academic as in the range of usefulness of the model (small deviation from plug flow, say Pe > 20) all conditions lead to res-... 

The Dispersion model has been widely used, especially for describing relatively small deviations from plug flow in packed beds and empty tubes. The availability of correlations that can be used to estimate D/uL for common reactor configurations makes this model especially convenient. Nevertheless, there are many situations, primarily high values of D/uL, for which the Dispersion model is not appropriate. Two alternative approaches to describing nonideal reactors are considered in the final sections of this chapter. 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

Advantages of three-phase fluidized beds over trickle beds and other fixed bed systems are temperature uniformity, high heat transfer, ability to add and remove catalyst particles continuously, and limited mass transfer resistances (both external to the particles and bubbles, because of turbulence and limited bubble size, and inside the particles owing to relatively small particle diameters). Disadvantages include substantial axial dispersion (of gas, liquid, and particles), causing substantial deviations from plug flow, and lack of predictability because of the complex hydrodynamics. There are two major applications of gas-liquid-solid-fluidized beds biochemical processes and hydrocarbon processing. 

The other two methods are subject to both these errors, since both the form ofi the RTD and the extent of micromixing are assumed. Their advantage is that they permit analytical solution for the conversion. In the axial-dispersion model the reactor is represented by allowing for axial diffusion in an otherwise ideal tubular-flow reactor. In this case the RTD for the actual reactor is used to calculate the best axial dififusivity for the model (Sec. 6-5), and this diffusivity is then employed to predict the conversion (Sec. 6-9). This is a good approximation for most tubular reactors with turbulent flow, since the deviations from plug-flow performance are small. In the third model the reactor is represented by a series of ideal stirred tanks of equal volume. Response data from the actual reactor are used to determine the number of tanks in series (Sec. 6-6). Then the conversion can be evaluated by the method for multiple stirred tanks in series (Sec. 6-10). 

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