Diffusion/dispersion model

Detachment constant A measure of the tendency of a complex to dissociate. Dew-point temperature The temperature at which saturation will occur. Diffusion/dispersion model A numerical model designed to simulate the atmo-... 

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. 

Dispersion model is based on Fick s diffusion law with an empirical dispersion coefficient substituted for the diffusion coefficient. The material balance is... 

Dispersion Model 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 represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

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. 

Mcllvried and Massoth [484] applied essentially the same approach as Hutchinson et al. [483] to both the contracting volume and diffusion-controlled models with normal and log—normal particle size distributions. They produced generalized plots of a against reduced time r (defined by t = kt/p) for various values of the standard deviation of the distribution, a (log—normal distribution) or the dispersion ratio, a/p (normal distribution with mean particle radius, p). 

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 ... 

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 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 ... 

Micromixing Models. Hydrodynamic models have intrinsic levels of micromixing. Examples include laminar flow with or without diffusion and the axial dispersion model. Predictions from such models are used directly without explicit concern for micromixing. The residence time distribution corresponding to the models could be associated with a range of micromixing, but this would be inconsistent with the physical model. 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

PeL can be used in place of Dh as the single parameter of the axial dispersion model. The physical interpretation of PeL is that it represents the ratio of the convective flux to fee diffusive (disposed) flux ... 

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. 

From this experimental setup, we expect detailed information how the influence of fluctuation has to be evaluated, how important it is under different diffusion conditions and how it can be taken into consideration in dispersion models. 

In another review, Hoffert discussed the social motivations for modeling air quality for predictive purposes and elucidated the components of a model. Meteorologic factors were summarized in terms of windfields and atmospheric stability as they are traditionally represented mathematically. The species-balance equation was discussed, and several solutions of the equation for constant-diffusion coefficient and concentrated sources were suggested. Gaussian plume and puff results were related to the problems of developing multiple-source urban-dispersion models. Numerical solutions and box models were then considered. The review concluded with a brief outline of the atmospheric chemical effects that influence the concentration of pollutants by transformation. 

Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

The dispersion model is one of the frequently used models. It describes the dispersion of the residence time of the phases according to Fig. 9.16, for example, in one-dimensional flows by superimposing the plug profile of the basic flow with a stochastic dispersion process in axial direction, which is constructed by analogy to Pick s first law of molecular diffusion ... 

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. 

As the alternative, a phenomenological description of turbulent mixing gives good results for many situations. An apparent diffusivity is defined so that a diffusion-type equation may be used, and the magnitude of this parameter is then found from experiment. The dispersion models lend themselves to relatively simple mathematical formulations, analogous to the classical methods for heat conduction and diffusion. 

Taylor (T2) and Westhaver (W5, W6, W7) have discussed the relationship between dispersion models. For laminar flow in round empty tubes, they showed that dispersion due to molecular diffusion and radial velocity variations may be represented by flow with a flat velocity profile equal to the actual mean velocity, u, and with an effective axial dispersion coefficient Djf = However, in the analysis, Taylor... 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

The Portland airshed dispersion model GRID (7.) is a conservation of mass, advectlon-diffusion code designed to perform... 

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. 

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