Axial dispersion experimental correlations

Wakao and Funazkri (1978) revealed that when mass transfer coefficients were measured by experiments involving adsorption or evaporation, the mass balance for the bed (see Chapter 6) should include a term accounting for axial dispersion. Previous correlations of experimental data were based upon a mass balance equation for the packed bed ignoring axial dispersion. It was shown that the mass transfer coefficient could be expressed in terms of the dimensionless Sherwood number (Sh) by the relation... 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

The importance of dispersion and its influence on flow pattern and conversion in homogeneous reactors has already been studied in Chapter 2. The role of dispersion, both axial and radial, in packed bed reactors will now be considered. A general account of the nature of dispersion in packed beds, together with details of experimental results and their correlation, has already been given in Volume 2, Chapter 4. Those features which have a significant effect on the behaviour of packed bed reactors will now be summarised. The equation for the material balance in a reactor will then be obtained for the case where plug flow conditions are modified by the effects of axial dispersion. Following this, the effect of simultaneous axial and radial dispersion on the non-isothermal operation of a packed bed reactor will be discussed. 

Fig. 10.16. Axial dispersion coefficients (solid line correlation, points CFD and experimental data).

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]... 

Figure 4-20 Correlation of experimental data (36 percent KATB feed) by the axial dispersion model.

As discussed in Chap. 3, there are a large number of models proposed to evaluate macromixing in a trickle-bed reactor. A brief summary of the reported experimental studies on the measurements of RTD in a cocurrent-downflow trickle-bed reactor is given in Table 6-7. Some of these experimental studies are described in more detail in a review by Ostergaard.94 Here we briefly review some of the correlations for the axial dispersion in gas and liquid phases based on these experimental studies. 

The experimental studies have shown that, in gas-liquid trickle-bed reactors, significant axial mixing occurs in both gas and liquid phases. When the RTD data are correlated by the single-parameter axial dispersion model, the axial dispersion coefficient (or Peclet number) for the gas phase is dependent upon both the liquid and gas flow rates and the size and nature of the packings. The axial dispersion coefficient for the liquid phase is dependent upon the liquid flow rate, liquid properties, and the nature and size of the packings, but it is essentially independent of the gas flow rate. 

Probably the simplest theoretical methods for determining the parameters from the experimental data involve the use of analytical solutions of simple column models and moment analysis, e.g. to determine the dead time or the Henry coefficient. In some cases where less accuracy is acceptable, parameters such as axial dispersion might be estimated by means of empirical correlations from literature. 

What models should be used either for scaleup or to correlate pilot plant data Section 9.1 gives the preferred models for nonisothermal reactions in packed beds. These models have a reasonable experimental basis even though they use empirical parameters D, hr, and Kr to account for the packing and the complexity of the flow field. For laminar flow in open tubes, use the methods in Chapter 8. For highly turbulent flows in open tubes (with reasonably large L/dt ratios) use the axial dispersion model in both the isothermal and nonisothermal cases. The assumption D = E will usually be safe, but do calculate how a PFR would perform. If there is a substantial difference between the PFR model and the axial dispersion model, understand the reason. For transitional flows, it is usually conservative to use the methods of Chapter 8 to calculate yields and selectivities but to assume turbulence for pressure drop calculations. 

Step 4. Use the experimental correlation to calculate the interpellet axial dispersion coefficient. 

Step 15. Determine the experimental correlation coefficient for interpellet axial dispersion via a conditional IF statement ... 

Analysis of experimental data for interpellet axial dispersion in packed beds has generated the following empirical correlation, as described by equation (22-84) and Table 22-6 ... 

In the calculation of the predicted response curves the axial dispersion coefficient and the external mass transfer coefficient were estimated from standard correlations and the effective pore diffusivily was determined from batch uptake rate measurements with the same adsorbent particles. The model equations were solved by orthogonal collocation and the computation time required for the collocation solution ( 20 s) was shown to be substantially shorter than the time required to obtain solutions of comparable accuracy by various other standard numerical methods. It is evident that the fit of the experimental breakthrough curves is good. Since all parameters were determined independently this provides good evidence that the model is essentially correct and demonstrates the feasibility of modeling the behavior of fairly complex multicomponent dynamic systems. 

The axial dispersion and tanks-in-series models are the two most common of models that have been developed as general semi-empirical correlations of mixing behavior, presumably bearing some relation to the actual flow pattern in the vessel. The model parameters have to be determined from experimental data and are then correlated as functions of fluid and flow properties and reactor configurations for use in design calculations. 

The axial dispersion coefficient can be determined experimentally, for instance, by finding the numerical value of the Peclet number, in impulse or step-response experiments. This is, however, possible, provided that experimental equipment (reactor) is available to carry out the experiments. This is not always the case. If no reactor experiments can be performed, the best approach is to estimate the value of the axial dispersion coefficient firom the available correlations. 

The above formulas are provided as theoretical guidance for the use of the dispersion model. For evaluation of actual coefficients the reader can consult the numerous experimental studies and correlations for tubes, packed and fluidized beds presented by Wen and Fan (58). One should remember that theory only justifies the use of the axial dispersion model at large Peclet nuu ers (Pe >> 1) and at small intensities of dispersion, i.e. D /uL < 0.15. Therefore, attempts in the literature to apply the dispersion model to small deviations from stirred tank behavior, i.e. for large intensities of dispersion, D /uL > 1, such as in describing liquid backmixing in bubble columns, should be considered with caution. Better physical models of the flow patterns are necessary for such situations and the dispersion model should be avoided. 

Because the parameter of the axial dispersion model, as observed from numerous experimental studies (58), has been so extensively correlated with Peclet number, designers consider the model useful for scaleup and use it for reactor calculations. The model gives a nice analytical expression for prediction of conversion of a single, irreversible first-order reaction (E(s) in Table 1 with Da replacing s). The expressions for exit concentrations for a system of reversible first-order reactions with the same axial dispersion coefficient (turbulent flow) are much more complex and their evaluation is computationally demanding. 

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