Correlations for Axial Dispersion

The following data apply to a specific experimental run. Bed voidage = 0.4, superficial velocity of fluid (based on an empty tube) = 1.2 cm/sec, and variances of output signals are found to be a = 39 sec and cr = 64 sec. Find D/wL. 

Bischoff and Levenspiel (1962) have shown that as long as the measurements are taken at least two or three particle diameters into the bed, then the open vessel boundary conditions hold closely. This is the case here because the measurements are made 15 cm into the bed. As a result this experiment corresponds to a one-shot input to an open vessel for which Eq. 12 holds. Thus 

Experiments show that the dispersion model well represents flow in packed beds and in pipes. Thus theory and experiment give lyiud for these vessels. We summarize them in the next three charts. 

Correlations similar to these are available or can be obtained for flow in beds of porous and/or adsorbing solids, in coiled tubes, in flexible channels, for pulsating flow, for non-Newtonians, and so on. These are given in Chapter 64 of Levenspiel (1996). 

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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)]. 

Figure 10.3 Correlation for axial dispersion coefficient In pipe flow, (Pe) versus (Re).

Correlations for axial dispersion coefficients in empty pipes and in packed beds are given in the Sections 4.10.6.3 and 4.10.6.4, respectively, so we can also calculate Dax without the need of an experiment (or prove the results of measurements, respectively). 

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

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Gas-solid fixed beds For axial dispersion in gas-solid fixed beds, the Edwards-Richardson correlation can be used (Wen and Fan, 1975 Andrigo et al., 1999). 

Figure 8-33. Correlation of axial dispersion coefficient for flow of fluids through pipes in laminar flow region (NRe < 2,000). (Source Wen, C. Y. and Fan, L. T, Models for Flow Systems and Chemical Reactors, Marcel Dekker Inc., 1975.)...

O. For liquids and gases, Ranz and Marshall correlation Nsh = - = 2.0 + 0.eNgNg AT dpVt uperP i-yRe R [E] Based on freely falling, evaporating spheres (see 5-20-C). Has been applied to packed beds, prediction is low compared to experimental data. Limit of 2.0 at low is too high. Not corrected for axial dispersion. [121][128] p. 214 [155] [110]... 

Correlate 20 gas studies and 16 liquid studies. Corrected for axial dispersion with ... 

Guedes de Carvalho et al. [20] studied axial mixing in slug flow. They found the following correlation for the dispersion coefficient ... 

Figure 5.8 (a) Correlation of axial dispersion coefficient for flow of fluids through pipes in... 

Krishnaswamy PR, Ganapathy R, Shemilt LW. Correlating parameters for axial dispersion in liquid fluidized systems. Can J Chem Eng 56 550-553, 1978. 

Even though the axial dispersion and tanks-in-series models discussed in this section can be used to handle a wide variety of nonideal flow situations, many reactors contain elements that do not satisfy the fundamental basis of a diffusionlike model and the development of consistent correlations for the dispersion coefficients might not be possible, hi these situations, more fundamental models, as presented in the previous sections, need to be utilized. 

Bubble columns. Tracers are used in bubble columns and gas-sparged slurry reactors mainly to determine the backmixing parameters of the liquid phase and/or gas-liquid or liquid-solid mass transfer parameters. They can be used for evaluation of holdup along the lines reviewed in the previous Section 6.2.1. However, there are simpler means of evaluating holdup in bubble columns, e.g. monitoring the difference in liquid level with gas and without gas flow. Numerous liquid phase tracer studies of backmixing have been conducted (132-149). Steady-state or continuous tracer inputs (132,134,140,142) as well as transient studies with pulse inputs (136,141,142,146) were used. Salts such as KC Jl or NaCil, sulfuric acid and dyes were employed as tracers. Electroconductivity detectors and spectrophotometers were used for tracer detection. The interpretation of results relied on the axial dispersion model. Various correlations for the dispersion... 

Chen and Wen (76) tried to improve on this model by allowing for axial dispersion of gas in the freeboard region and by means of assumptions which give more realistic profiles of particle holdup in the dilute phase. Entrainment is calculated from their recent correlation (77) which leads to an exponentially decreasing solids mass flux and makes allowance for solids returning to the bed surface along the vessel walls. Ejected particles were... 

Interphase Mass Transfer. There are a number of interphase mass transfer steps that must occur in a trickle flow reactor. The mass transfer resistances can be considered as occurring at the more or less stagnant fluid layer interfaces, i.e., on the gas and/or the liquid side of the gas/llquld Interface and on the liquid side of the liquid/solid Interface. The mass transfer correlations (8) indicate that the gas/llquld Interface and the liquid/solid interface mass transfer resistances decrease with higher liquid velocity and smaller particle size. Thus, in the PDU, the use of small inert particles partially offsets the adverse effect of low velocity. These correlations indicate that for this system, external mass transfer limitations are more likely to occur in the PDU than in the commercial reactor because of the lower liquid velocity, but that probably there is no limitation in either. If a mass transfer limitation were present, it would limit conversion in a way similar to that shown for axial dispersion and incomplete catalyst wetting illustrated in Figure 1. Due to the uncertainty in the correlations and in the physical properties of these systems, particularly the molecular diffuslvities, it is of interest to examine if external mass transfer is influencing the PDU results. 

In chemical reaction engineering single phase reactors are often modeled by a set of simplified ID heat and species mass balances. In these cases the axial velocity profile can be prescribed or calculate from the continuity equation. The reactor pressure is frequently assumed constant or calculated from simple relations deduced from the area averaged momentum equation. For gases the density is normally calculated from the ideal gas law. Moreover, in situations where the velocity profile is neither flat nor ideal the effects of radial convective mixing have been lumped into the dispersion coefficient. With these model simplifications the semi-empirical correlations for the dispersion coefficients will be system- and scale specific and far from general. 

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