Packed beds dispersion

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. 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 reaclor with a small deviation from phig ffow, without specifying the magnitude ol 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. 

FIG. 16-9 General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to mimhered paragraphs in the text 1, pore diffusion 2, solid diffusion 3, reaction kinetics at phase boundary 4, external mass transfer 5, fluid mixing. 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

Correlations for axial dispersion in beds packed with very small particles (<50 Im) that take into account the holdup of liquid in a bed are discussed by Horvath and Lin [J. Chromatogr, 126, 401 (1976)]. 

TABLE 16-10 Coefficients for Axial Dispersion Correlations in Packed Beds Based on Eq (16-79)... 

FIG. 16 11 Axial dispersion coefficient correlations for well-packed beds for e = 0.4. 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

Peclet number for dispersion Pe = uUD where t/ is a Bnear velocity, L is a hnear dimension, and is the dispersion coefficient. In packed beds, Pe = udp/De, where u is the interstitial velocity and dp is the pellet diameter. 

Re = R nolds number, dpS UolV Sc = Schmidt number, V/D D = axial dispersion coefficient dp = Diameter of particle or empty tube = Fraction voids in packed bed Uq = Superficial velocity in the vessel. 

Figure 3.2.1 illustrates the mixing in packed beds (Wilhelm 1962). As Reynolds number approaches the industrial range Rep > 100, the Peclet numbers approach a constant value. This means that dispersion is influenced by turbulence and the effect of molecular diffusion is negligible. 

The original Rate Theory which describes dispersion in packed beds evolved over a number of years, probably starting with the work of Lapidus and Amundson [6] in 1952, extended by that of Glueckauf [7] and Tunitski [8] in 1954. The final form of the equation that described dispersion in packed beds as a function of the linear... 

To identify the pertinent HETP equation that describes dispersion in a packed bed, the following logical procedure will require to be carried out. 

The dispersion model has been successfully employed in modeling the behavior of packed bed reactors. In this case. 

The Peclet numbers decrease when the dispersion coefficients increase. In the Reynolds number range of 10-200, in a packed bed of pellets, Peii = 2 and PeH = 0.5 (119, 120). The dispersions in the transverse... 

Further work regarding the axial dispersion of gas in irrigated packed beds seems needed, and it may be noted, with particular regard to gas-liquid-particle processes, that no results have been reported for beds of cylindrically or spherically shaped packing materials. 

Glaser and Lichtenstein (G3) measured the liquid residence-time distribution for cocurrent downward flow of gas and liquid in columns of -in., 2-in., and 1-ft diameter packed with porous or nonporous -pg-in. or -in. cylindrical packings. The fluid media were an aqueous calcium chloride solution and air in one series of experiments and kerosene and hydrogen in another. Pulses of radioactive tracer (carbon-12, phosphorous-32, or rubi-dium-86) were injected outside the column, and the effluent concentration measured by Geiger counter. Axial dispersion was characterized by variability (defined as the standard deviation of residence time divided by the average residence time), and corrections for end effects were included in the analysis. The experiments indicate no effect of bed diameter upon variability. For a packed bed of porous particles, variability was found to consist of three components (1) Variability due to bulk flow through the bed... 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

Chapters 8 and Section 9.1 gave preferred models for laminar flow and packed-bed reactors. The axial dispersion model can also be used for these reactors but is generally less accurate. Proper roles for the axial dispersion model are the following. 

Correlations for E are not widely available. The more accurate model given in Section 9.1 is preferred for nonisothermal reactions in packed-beds. However, as discussed previously, this model degenerates to piston flow for an adiabatic reaction. The nonisothermal axial dispersion model is a conservative design methodology available for adiabatic reactions in packed beds and for nonisothermal reactions in turbulent pipeline flows. The fact that E >D provides some basis for estimating E. Recognize that the axial dispersion model is a correction to what would otherwise be treated as piston flow. Thus, even setting E=D should improve the accuracy of the predictions. 

FIGURE 9.11 Comparison of piston flow and axial dispersion for the packed-bed reactor of Example 9.6 T, = r.,.,H = 373K. 

The global design equations for packed beds—e.g.. Equations (10.1), (10.9), (10.39), and (10.40)—all have a similar limitation to that of the axial dispersion model treated in Chapter 9. They all assume steady-state operation. Adding an accumulation term, da/dt accounts for the change in the gas-phase inventory of component A but not for the surface inventory of A in the adsorbed form. The adsorbed inventory can be a large multiple of the gas-phase inventory. 

Compare Equation (11.42) with Equation (9.1). The standard model for a two-phase, packed-bed reactor is a PDE that allows for radial dispersion. Most trickle-bed reactors have large diameters and operate adiabaticaUy so that radial gradients do not arise. They are thus governed by ODEs. If a mixing term is required, the axial dispersion model can be used for one or both of the phases. See Equations (11.33) and (11.34). 

Most biochemical reactors operate with dilute reactants so that they are nearly isothermal. This means that the packed-bed model of Section 9.1 is equivalent to piston flow. The axial dispersion model of Section 9.3 can be applied, but the correction to piston flow is usually small and requires a numerical solution if Michaehs-Menten kinetics are assumed. 

Radial dispersion coefficient for heat in a packed-bed 9.3 Axial dispersion coefficient for temperature in PDE Sec. 9.1 model... 

However, in contrast to the two classes of dispersive mixers mentioned before, the attached flow-through channel contains a packed bed of particles which may carry a catalyst. This chamber is much larger than the typical dimensions of the inlet channels (e.g. compare with Section 5.1.2). The packed bed and its interstices influence the gas/liquid flow patterns, e.g. a trickle-bed operation may be established. 

A great savings in enzyme consumption can be achieved by immobilizing the enzyme in the reactor (Fig. 12). In addition to the smaller amount of enzyme required, immobilization often increases the stability of the enzyme. Several designs of immobiliz-ed-enzyme reactors (lERs) have been reported, with open-tubular and packed-bed being the most popular. Open-tubular reactors offer low dispersion but have a relatively small surface area for enzyme attachment. Packed-bed reactors provide extremely high surface areas and improved mass transport at the cost of more dispersion. 

J. Gotz, K. Zick, C. Heinen, T. Konig 2002, (Visualisation of flow processes in packed beds with NMR imaging Determination of the local porosity, velocity vector and local dispersion coefficients), Chem. Eng. Process. 41 (7), 611-630. 

The modeling of mass transport in packed bed reactors applies the theory of dispersion [32]. The conservation of mass for the average concentration [Pg.515]

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