The Dispersion Model

Models are useful for representing flow in real vessels, for scale up, and for diagnosing poor flow. We have different kinds of models depending on whether flow is close to plug, mixed, or somewhere in between. 

Chapters 13 and 14 deal primarily with small deviations from plug flow. There are two models for this the dispersion model and the tanks-in-series model. Use the one that is comfortable for you. They are roughly equivalent. These models apply to turbulent flow in pipes, laminar flow in very long tubes, flow in packed beds, shaft kilns, long channels, screw conveyers, etc. 

For laminar flow in short tubes or laminar flow of viscous materials these models may not apply, and it may be that the parabolic velocity profile is the main cause of deviation from plug flow. We treat this situation, called the pure convection model, in Chapter 15. 

If you are unsure which model to use go to the chart at the beginning of Chapter 15. It will tell you which model should be used to represent your setup. 

If the concentration of the tracer molecules in the reactor effluent are known, then the measured response will depend on the length of the reactor, the rate of diffusion, and mean fluid velocity. The response 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if the Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m2/sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Flow patterns in the reactor can vary greatly. To characterize backmixing, of the longitudinal dispersion number, D/uL, is often used, 

In a packed bed or flow in pipes, the dispersion number is also defined as D/ud, where d is the particle size in packed beds or the tube diameter in empty pipes. 

Fick s diffusion law is used to describe dispersion. In a tubular reactor, either empty or packed, the depletion of the reactant and non-uniform flow velocity profiles result in concentration gradients, and thus dispersion in both axial and radial directions. Fick s law for molecular diffusion in the x-direction is defined by 

In the PFR, reactant concentration will fall along the reactor length in proportion to the amount of reaction occurring. The axial concentration 

An unsteady state mass balance over the typical reactor element lying between x and x + 5x gives rise to the partial differential equation 

This assumes that the concentration at any value of x is not a function of radius. Ca is the concentration of reacting species A, u the mean convective velocity, which is assumed to be neither a function of axial or radial position, and Ta is the reaction rate of A based on unit volume. If nth-order power law kinetics pertain, i.e. 

The only two parameters appearing in eqn. (65) are the dispersion number, DjiiL, or inverse Peclet number, and the Damkohler number, or dimensionless rate group, t/jCa, - Solutions to eqn. (65) are therefore functions only of these two groups. If term (4) in eqn. (65) is absent, then 

The Peclet number, uLjD, when written in the form (L /D)l(Llu) is seen to be a ratio of characteristic dispersion time to characteristic residence time and the Damkohler number can, in similar manner, be considered as a ratio of characteristic residence time, L/u, to characteristic reaction time, l/feCA [59]. 

The Dispersion model, or dispersed plug-flow model, is an extension of the ideal plug-flow model that allows for some mixing in the direction of flow. The Dispersion model is one dimensional. Like the ideal plug-flow model, all of the concentration and temperature gradients are in the direction of flow. There are no concentration or temperature gradients normal to the direction of flow. 

If the packing is porous, e.g., porous catalyst particles, the volume of the pores is not included in ei. 

The fraction of the reactor that is occupied by the packing is 1 — i. In other words, 1 — i, is the geometrical volume of the packing per unit geometrical volume of reactor. 

We will assume that the bed is isotropic. Therefore, the fraction of the cross-sectional area that is occupied by the interstices also is Ci, and the cross-sectional area through which fluid actually flows is ejAc. Similarly, the volume of reactor through which fluid actually flows is tyAcL (=fiiV). 

In this equation, z is the dimension in the direction of flow, i.e., the axial direction in a tubular reactor. As usual, v is the volumetric flow rate. We will assume that the density of the fluid flowing through the reactor is constant, so that v is the same at every cross section along the length of the reactor. Finally, —is the rate of disappearance of A per unit of geometric reactor volume. 

An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically the process can be described by Pick s equation with a dispersion coefficient instead of a diffusion coefficient. The dispersion coefficient D is associated with a linear dimension L and a linear velocity u in the Peclet number, Pe = uL/D. 

A significant merit of the dispersion model is some experimental correlations for the Peclet number. There are no such direct correlations for the parameters of the Gamma or Gaussian or other similar models, 

In a vessel with dispersion in the axial direction, the steady state equation for a reaction of order n is 

The derivation and other forms of the dispersion-reaction equation are in problem P5.08.01. The solution of the partial differential Equation (b) for tracer operation is recorded in the Literature and is quoted in problem P5.08.04. 

Two different sets of boundary conditions are applicable to these differential equations. The corresponding operating conditions are shown on the figure with problem 5.08.01 In normal operation with closed ends , reactant is brought in by bulk flow and carried away by both bulk and dispersion flow, so at the inlet where L = 0 or z = 0, 

Conversion with known flow patterns. Laminar flow 557 

Vessels in which chemical reactions are conducted in the plant or laboratory are of various shapes and internal arrangements. The distribution of residence times in them of various reacting molecules or aggregates, the RTD, is a key datum for determining the performance of a reactor, either the expected conversion or the range in which the conversion must fall. How the RTD is measured or calculated and applied is the subject of this chapter. The main application of interest here is to find how nearly a particular vessel approaches some standard ideal behavior, or what its efficiency is. 

A number of special terms are defined in the Glossary, Table 5.1. Equations for tracer response functions are summarized in Table 5.2. 

Primarily non-reactive substances that can be easily analyzed for concentration are used as tracers. When making a test, tracer is injected to 

Quantitatively, the efficiency at a specified conversion level, x, is defined as the ratio of the mean residence time or reactor volume in a plug flow reactor (PFR) to that of the reactor in question, 

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The dispersion model is typically used to determine the downwind concentrations of released materials and the total area affected. Two models are available the plume and the puff. The plume describes continuous releases the puff describes instantaneous releases. 

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. 23-15 Chemical conversion by the dispersion model, (a) First-order reaction, volume relative to plug flow against residual concentration ratio, (h) Second-order reaction, residual concentration ratio against kC t. 

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

Levenspiel and Bisehoff [24] eompared the fraetion unreaeted by the dispersion model to the solution with that for plug flow ... 

It should be noted that the dispersion model for radioactive material developed in WASH-1400 for reactor sites as a class cannot be applied to individual sites without significant refinement and sensitivity tests,... 

Many mixing models which utilize the simplified concepts of micro-mixing and segregation have been introduced. Most notable of these are the two-environment models of Chen and Fan (19), Kearns and Manning (20), and others (21, 22), and the dispersion models of Spielman and Levenspiel (23), and Kattan and Adler (24). 

Letting the element distance AZ approach zero in the finite-difference form of the dispersion model, gives... 

The dispersion model of example DISRE is extended for non-isothermal reactions to include the dispersion of heat from a first-order reaction. 

The dispersion model approach was first proposed to simulate dynamic absorption processes [49], The dispersion model assumes that the small intestine can be considered as a uniform tube with constant axial velocity, constant dispersion behavior, and uniform concentration across the tube diameter. Then the absorption of highly soluble drugs in the small intestine can be delineated by the following dispersion model equation ... 

Numerical solutions to equation 11.2.9 have been obtained for reaction orders other than unity. Figure 11.11 summarizes the results obtained by Levenspiel and Bischoff (18) for second-order kinetics. Like the chart for first-order kinetics, it is most appropriate for use when the dimensionless dispersion group is small. Fan and Bailie (19) have solved the equations for quarter-order, half-order, second-order, and third-order kinetics. Others have used perturbation methods to arrive at analogous results for the dispersion model (e.g. 20,21). 

ILLUSTRATION 11.6 USE OF THE DISPERSION MODEL TO DETERMINE THE CONVERSION LEVEL OBTAINED IN A NONIDEAL FLOW REACTOR... 

Figure 5-6 The initial acceleration and buoyancy of the released material affects the plume character. The dispersion models discussed in this chapter represent only ambient turbulence. Adapted from Steven R. Hanna and Peter J. Drivas, Guidelines for Use of Vapor Cloud Dispersion Models (New York American Institute of Chemical Engineers, 1987), p. 6.

The coordinate system used for the dispersion models is shown in Figures 5-7 and 5-8. The x axis is the centerline directly downwind from the release point and is rotated for different wind directions. The y axis is the distance off the centerline, and the z axis is the elevation... 

Packed beds usually deviate substantially from plug flow behavior. The dispersion model and some combinations of PFRs and CSTRs or of multiple CSTRs in series may approximate their behavior. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some correlations of the Peclet number that have been achieved are cited in problem P5.08.14. It is related to the variance of an RTD, as discussed in problem P5.08.04. Consequently the dispersion model and the Gamma or Gaussian are interrelated. 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

The input value of the dispersion model represents the odour load as established by a psychophysical experiment. The experiment relates a human sensation to a physical quantity. The human sensation to be recorded usually encompasses questions concerning the detectability, the intensity or the quality of the odour. In the case of odour pollution, the physical quantity is often expressed as the dilution number or the relative concentration. 

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