Dispersion model fluids

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. 

Dispersion modeling equations for water systems take the same form as those presented later in this chapter for the atmosphere. Analytical solutions tire not nearly as complicated or difficult, since the bulk motion of the fluid (in this case, wtiicr) is a weak vtiriablc with respect to m.ignitude, direction, lime, and position as it is when the fluid is air. 

The maximum-mixedness model (MMM) for a reactor represents the micromixing condition of complete dispersion, where fluid elements mix completely at the molecular level. The model is represented as a PFR with fluid (feed) entering continuously incrementally along the length of the reactor, as illustrated in Figure 20.1 (after Zwieter-ing, 1959). The introduction of feed incrementally in a PFR implies complete mixing... 

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 CD model was first proposed by Curl (1963) to describe coalescence and breakage of a dispersed two-fluid system. In each mixing event, two fluid particles with distinct compositions first coalesce and then disperse with identical compositions.75 Written in terms of the two compositions (f>A and [Pg.292]

Farrell, J. A., J. Murlis, X. Long, W. Li, and R. T. Card6. Filament-based atmospheric dispersion model to achieve short time scale structure of odor plumes. Environ. Fluid Mech. 2, 143-169 (2002). 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

Equation (I-l) is the general representation of the dispersion model. The dispersion coefficient is a function of both the fluid properties and the flow situation the former have a major effect at low flow rates, but almost none at high rates. In this general representation, the dispersion coefficient and the fluid velocity are all functions of position. The dispersion coefficient, D, is also in general nonisotropic. In other words, it has different values in different directions. Thus, the coefficient may be represented by a second-order tensor, and if the principal axes are taken to correspond with the coordinate system, the tensor will consist of only diagonal elements. 

First Moments. For both of the dispersed plug-flow cases Mi = 0. This means that the center of gravity of the solute moves with the mean speed of the flowing fluid. For the uniform and the general dispersion models, however, this is not always true. If the solute concentration is initially uniform over a cross-sectional plane, it can be shown (A6) that... 

When the gross flow pattern of fluid deviates greatly from plug flow because of channeling or recirculation of fluid, eddies in odd corners, etc., then the dispersion model or the tanks-in-series model can not satisfactorily characterize flow in the vessel. This type of flow can be found,... 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

Observations on fixed-bed dispersion models The role of the interstitial fluid (with S. Sundar-esan and N.R. Amundson). AJ.Ch.EJ. 26,529-536 (1980). 

Scanning electron microscopy/energy dispersive X-ray (SEM/EDX) analyses have been used to probe the migration of vanadium in model fluid cracking catalysts (FCC). At the experimental conditions used, vanadium can migrate either from a Eu3+-exchanged Y (EuY) zeolite to an AAA-alumina gel or vice versa, depending on the type of vanadium precursor used. 

Consequence modeling, for the purposes of the illustrations given in this chapter, means the prediction of ambient atmospheric concentrations using models for quantifying the release of fluids from containment, and the formation of vapor and liquid aerosol plumes using dispersion models. 

The two-phase theory of fluidization has been extensively used to describe fluidization (e.g., see Kunii and Levenspiel, Fluidization Engineering, 2d ed., Wiley, 1990). The fluidized bed is assumed to contain a bubble and an emulsion phase. The bubble phase may be modeled by a plug flow (or dispersion) model, and the emulsion phase is assumed to be well mixed and may be modeled as a CSTR. Correlations for the size of the bubbles and the heat and mass transport from the bubbles to the emulsion phase are available in Sec. 17 of this Handbook and in textbooks on the subject. Davidson and Harrison (Fluidization, 2d ed., Academic Press, 1985), Geldart (Gas Fluidization Technology, Wiley, 1986), Kunii and Levenspiel (Fluidization Engineering, Wiley, 1969), and Zenz (Fluidization and Fluid-Particle Systems, Pemm-Corp Publications, 1989) are good reference books. 

The dimensionless term (9/u0 L, where 9 is the axial dispersion coefficient, u0 is the superficial fluid velocity, and L is the expanded-bed height) is the column-vessel dispersion number, Tc, and is the inverse of the Peclet number of the system. Two limiting cases can be identified from the axial dispersion model. First, when 9/u0L - 0, no axial dispersion occurs, while when 9/u0 L - 00 an infinite diffusivity is obtained and a stirred tank performance is achieved. The dimensionless term Fc, can thus be utilized as an important indicator of the flow characteristics within a fluidized-bed system.446... 

B. Oesterle and L. L Zaichik. On lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows. Phys. Fluids, 16(9) 3374-3384, 2004. 

So far, only the axial dispersion model has been used for scaleup purposes. Very little knowledge on the effects of reactor configuration and flow conditions on the parameters of more complex macromixing models (e.g., the two-parameters model, etc.) is available. Since these complex models are more realistic, more information on the relation between their parameters and the system conditions, such as packing size, fluid properties, and flow rates, needs to be obtained. At present, complex models are not very useful for scaleup purposes. 

Actually, we are quite confident that the temperatures in the older reactor were accurate, at least to about db5°C, and probably as accurate as those in the annular reactor. A more likely source of the discrepancy lies in the undoubted existence of some back-mixing or partial gas bypassing in the older, fluid-bed reactor. Although, the a priori applicability of the model seemed doubtful, we have attempted to estimate the magnitude of this back-mixing effect by using the dispersion model described by Levenspiel (14). 

Here we use a single parameter to account for the nonideality of our reactor. This parameter is most always evaluated by analyzing the RTD determined from a tracer test. Examples of one-parameter models for a nonideal CSTR include the reactor dead volume V, where no reaction takes place, or the fraction / of fluid bypassing the reactor, thereby exiting unreacted. Examples of one-parameter models for tubular reactors include the tanks-in-series model and the dispersion model. For the tanks-in-series model, the parameter is the number of tanks, n, and for the dispersion model, it is the dispersion coefficient D,. Knowing the parameter values, we then proceed to determine the conversion and/or effluent concentrations for the reactor. 

Harada and co-workers (HI) developed two coalescence-redispersion models to describe micromixing in a continuous-flow reactor. In the first model, the incoming dispersed-phase fluid is assumed to consist of uniformly sized droplets. These droplets undergo 0 to n coalescences and redispersions with surrounding droplets of a constant average concentra-... 

Early attempts to approximate gas-solid contacting in fluid catalyst beds were based on the assumption either of isothermal plug flow of the fluidizing gas through the bed with the catalyst uniformly distributed or of isothermal complete mixing of the gas within the bed. The simple dispersion model, falling between the above two cases, was also used (G8, R4). Evidence from both large-scale and laboratory observations (G9a, L12),... 

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