Axial flow model

Bauer, E. G., G. R. Houdayer, and H. M. Sureau, 1978, A Nonequilibrium Axial Flow Model and Application to Loss-of-Coolant Accident, in Proc. Transient Two-Phase Flow CSN1 Specialists Meeting, 1976, Atomic Energy of Canada 7 429-437. (3)... 

Bauer, E. G., Houdayer, G. R. Sureau, H. M. 1976. A nonequilibrium axial flow model and application to loss-of-coolant accident analysis The CLYSTERE system code. Proceedings of CSNI Meeting on Transient Two-Phase Flow, Toronto, Canada. 

The simple axial flow model, which undoubtedly represents the best utilization of the internal volume of the converter, has two drawbacks. To contain pressure drops, it is imperative to use relatively large-size catalysts, which are less efficient due to pore diffusion limitation, and this results in an increase in the specific volume of the converter. Despite this technical concession, low bed-height to bed-diameter ratios are also required, leading to large diameters and consequent wall thicknesses of the vessel. Such converters become relatively expensive, and heavy. Several alternative flow models, taking advantage of a larger bed cross-sectional area in proportion to the vessel cross-section, have been found to overcome this problem. 

Figure 6.1. Flow models in adiabatic catalyst beds. Besides the simple axial-flow model, several alternative flow-models have been proposed to contain the pressure drop when relatively small-size catalysts, which are more active, are used.

A dense-bed center-fed column (Fig. 22-li) having provision for internal crystal formation and variable reflux was tested by Moyers et al. (op. cit.). In the theoretical development (ibid.) a nonadiabatic, plug-flow axial-dispersion model was employed to describe the performance of the entire column. Terms describing interphase transport of impurity between adhering and free liquid are not considered. 

Figure 7-11. Apparatus for testing axial-flow cascade model.

This model is referred to as the axial dispersed plug flow model or the longitudinal dispersed plug flow model. (Dg)j. ean be negleeted relative to (Dg)[ when the ratio of eolumn diameter to length is very small and the flow is in the turbulent regime. This model is widely used for ehemieal reaetors and other eontaeting deviees. 

The axial dispersion model also gives a good representation of fluid mixing in paeked-bed reaetors. Figure 8-34 depiets the eorrelation for flow of fluids in paeked beds. 

Comparison of solutions of the axially dispersed plug flow model for different boundary conditions... 

The axial dispersion plug flow model is used to determine the performanee of a reaetor with non-ideal flow. Consider a steady state reaeting speeies A, under isothermal operation for a system at eonstant density Equation 8-121 reduees to a seeond order differential equation ... 

Both the tank in series (TIS) and the dispersion plug flow (DPF) models require traeer tests for their aeeurate determination. However, the TIS model is relatively simple mathematieally and thus ean be used with any kineties. Also, it ean be extended to any eonfiguration of eompartments witli or without reeycle. The DPF axial dispersion model is eomplex and therefore gives signifieantly different results for different ehoiees of boundary eonditions. 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

The two models commonly used for the analysis of processes in which axial mixing is of importance are (1) the series of perfectly mixed stages and (2) the axial-dispersion model. The latter, which will be used in the following, is based on the assumption that a diffusion process in the flow direction is superimposed upon the net flow. This model has been widely used for the analysis of single-phase flow systems, and its use for a continuous phase in a two-phase system appears justified. For a dispersed phase (for example, a bubble phase) in a two-phase system, as discussed by Miyauchi and Vermeulen, the model is applicable if all of the dispersed phase at a given level in a column is at the same concentration. Such will be the case if the bubbles coalesce and break up rapidly. However, the model is probably a useful approximation even if this condition is not fulfilled. It is assumed in the following that the model is applicable for a continuous as well as for a dispersed phase in gas-liquid-particle operations. 

This section has based scaleups on pressure drops and temperature driving forces. Any consideration of mixing, and particularly the closeness of approach to piston flow, has been ignored. Scaleup factors for the extent of mixing in a tubular reactor are discussed in Chapters 8 and 9. If the flow is turbulent and if the Reynolds number increases upon scaleup (as is normal), and if the length-to-diameter ratio does not decrease upon scaleup, then the reactor will approach piston flow more closely upon scaleup. Substantiation for this statement can be found by applying the axial dispersion model discussed in Section 9.3. All the scaleups discussed in Examples 5.10-5.13 should be reasonable from a mixing viewpoint since the scaled-up reactors will approach piston flow more closely. 

These boundary conditions are really quite marvelous. Equation (9.16) predicts a discontinuity in concentration at the inlet to the reactor so that ain a Q+) if D >0. This may seem counterintuitive until the behavior of a CSTR is recalled. At the inlet to a CSTR, the concentration goes immediately from to The axial dispersion model behaves as a CSTR in the limit as T) — 00. It behaves as a piston flow reactor, which has no inlet discontinuity, when D = 0. For intermediate values of D, an inlet discontinuity in concentrations exists but is intermediate in size. The concentration n(O-l-) results from backmixing between entering material and material downstream in the reactor. For a reactant, a(O-l-) [Pg.332]

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. 

Laminar Pipeline Flows. The axial dispersion model can be used for laminar flow reactors if the reactor is so long that At/R > 0.125. With this high value for the initial radial position of a molecule becomes unimportant. 

The molecule diffuses across the tube and samples many streamlines, some with high velocity and some with low velocity, during its stay in the reactor. It will travel with an average velocity near u and will emerge from the long reactor with a residence time close to F. The axial dispersion model is a reasonable approximation for overall dispersion in a long, laminar flow reactor. The appropriate value for D is known from theory ... 

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. 

Example 9.6 Compare the nonisothermal axial dispersion model with piston flow for a first-order reaction in turbulent pipeline flow with Re= 10,000. Pick the reaction parameters so that the reactor is at or near a region of thermal runaway. 

Solution The axial dispersion model requires the simultaneous solution of Equations (9.14) and (9.24). Piston flow is governed by the same equations except that D = E = Q. The following parameter values give rise to a near runaway ... 

FIGURE 9.12 Comparison of piston flow and axial dispersion models at conditions near thermal... 

A more dramatic comparison of the piston flow and axial dispersion models is shown in Figure 9.12. Input parameters are the same as for Figure 9.11 except that Tin and T aii were increased by 1K. This is another example of parametric sensitivity. Compare Example 9.2. 

Observe that the axial dispersion model provides a lower and thus more conservative estimate of conversion than does the piston flow model given the same values for the input parameters. There is a more subtle possibility. The model may show that it is possible to operate with less conservative values for some parameters—e.g., higher values for Tin and T aii— without provoking adverse side reactions. 

Water at room temperature is flowing through a 1.0-in i.d. tubular reactor at Re= 1000. What is the minimum tube length needed for the axial dispersion model to provide a reasonable estimate of reactor performance What is the Peclet number at this minimum tube length Why would anyone build such a reactor ... 

These simple situations can be embellished. For example, the axial dispersion model can be applied to the piston flow elements. However, uncertainties in reaction rates and mass transfer coefficients are likely to mask secondary effects such as axial dispersion. 

Axial Dispersion. Enthusiastic modelers sometimes add axial dispersion terms to their two-phase, piston flow models. The component balances are... 

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. 

Washout experiments can be used to measure the residence time distribution in continuous-flow systems. A good step change must be made at the reactor inlet. The concentration of tracer molecules leaving the system must be accurately measured at the outlet. If the tracer has a background concentration, it is subtracted from the experimental measurements. The flow properties of the tracer molecules must be similar to those of the reactant molecules. It is usually possible to meet these requirements in practice. The major theoretical requirement is that the inlet and outlet streams have unidirectional flows so that molecules that once enter the system stay in until they exit, never to return. Systems with unidirectional inlet and outlet streams are closed in the sense of the axial dispersion model i.e., Di = D ut = 0- See Sections 9.3.1 and 15.2.2. Most systems of chemical engineering importance are closed to a reasonable approximation. 

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