The Axial Dispersion Model

Here q is the heat conduction due to the temperature gradient. The simplest way to express this is using Fourier s2 law 

2Jean Baptiste Joseph Fourier, French mathematician, 1768 - 1830 

Needless to say, the assumption of plug flow is not always appropriate. In plug flow we assume that the convective flow, i. e., the flow at velocity qjAt = v that is caused by a compressor or pump, is dominating any other transport mode. In practice this is not always so and dispersion of mass and heat, driven by concentration and temperature gradients are sometimes significant enough to need to be included in the model. We will discuss such a model in detail, not only because of its importance, but also because the techniques used to handle the ensuing boundary value differential equations are similar to those used for other diffusion-reaction problems such as catalyst pellets, as well as for counter-current processes. 

A simple correction to piston fiow is to add an axial diffusion term. The resulting equation remains an ODE and is known as the axial dispersion model  

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

At lower Reynolds numbers, the axial velocity profile will not be flat and it might seem that another correction must be added to Equation (9.14). It turns out, however, that Equation (9.14) remains a good model for real turbulent reactors (and even some laminar ones) given suitable values for D. The model lumps the combined effects of fluctuating velocity components, nonflat velocity profiles, and molecular diffusion into the single parameter D. 

FIGURE 9.6 Peclet number, Pe = udt/D, versus Reynolds number, Re = pd,u/(jl, for flow in an open tube. 

FIGURE 9.7 Peclet number Pe = usdp /D, versus Reynolds number, Re — pdpu, / p for packed beds. 

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Two alternative approaches are used ia axial mixing calculations. For differential contactors, the axial dispersion model is used, based on an equation analogous to equation 13 ... 

Langmuir [18] first proposed the axial dispersion model and obtained steady state solutions from die following boundary eonditions ... 

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. 

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. 

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 axial dispersion model has a long and honored history within chemical engineering. It was first used by Langmuir, who also used the correct boundary... 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

FIGURE 9.8 The axial dispersion model applied to a closed system. 

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. 

Adiabatic Reactors. Like isothermal reactors, adiabatic reactors with a flat velocity profile will have no radial gradients in temperature or composition. There are axial gradients, and the axial dispersion model, including its extension to temperature in Section 9.4, can account for axial mixing. As a practical matter, it is difficult to build a small adiabatic reactor. Wall temperatures must be controlled to simulate the adiabatic temperature profile in the reactor, and guard heaters may be needed at the inlet and outlet to avoid losses by radiation. Even so, it is hkely that uncertainties in the temperature profile will mask the relatively small effects of axial dispersion. 

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 ... 

As seen in Chapter 8, the stability criterion becomes quite demanding when At is large. The axial dispersion model may then be a useful alternative... 

The axial dispersion model is readily extended to nonisothermal reactors. The turbulent mixing that leads to flat concentration profiles will also give flat temperature profiles. An expression for the axial dispersion of heat can be written in direct analogy to Equation (9.14) ... 

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. 

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 ... 

Turn now to the axial dispersion model. Plausible values for the dispersion coefficients at Re = 10,000 are... 

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. 

Determine the yield of a second-order reaction in an isothermal tubular reactor governed by the axial dispersion model with Pe = 16 and kt = 2. 

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 ... 

Chapter 15 provides additional discussion of the axial dispersion model and of methods for measuring dispersion coefficients. A more advanced account is given in... 

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. 

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. 

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. 

The axial dispersion model discussed in Section 9.3 is a simplified version of Equation (14.19). Analytical solutions for unsteady axial dispersion are given in Chapter 15. 

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. 

FIGURE 15.10 Transient response measurements for systems governed by the axial dispersion model (a) closed system (b) open system. 

Solution The first step in the solution is to find a residence time function for the axial dispersion model. Either W t) or f(t) would do. The function has Pe as a parameter. The methods of Section 15.1.2 could then be used to determine which will give the desired relationship between Pe and... 

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