199 dispersion coefficient

However, this model is a disaster at predicting how much the plume spreads. From observation, we know that it actually has spread about 1 kilometer. From the arguments in Section 2.4, we know that the width of this peak I should be about 

In gases, diffusion coefficients are about 0.1 cm /sec, and the time is about 10 km/(15 km/ hr), or 40 minutes. On this basis, / should be about 30 centimeters, 3,000 times less than the observed width of 1 kilometer. A factor of 3,000 is a big error, even for engineers. 

The explanation for this major discrepancy is the wind. In previous chapters, mixing occurred by diffusion caused by molecular motion. Here, mixing occurs as the wind blows the plume over woods, around hills, and across lakes. This mixing is more rapid than diffusion because of the flow. 

We now are in something of a quandary. We have a good diffusion model in Eq. 4.1-1 that explains most of the qualitative features of the plume, but this model grossly underpredicts the effects. To resolve this, we assume that mass transport in the plume is described by the flux equation 

The new dispersion coefficient must usually be measured experimentally. Like the diffusion coefficient, the dispersion coefficient has dimensions of (L lt). Unlike the diffusion coefficient, the dispersion coefficient is largely independent of chemistry. It will not be a strong function of molecular weight or chemical structure, but will have close to the same values for carbon monoxide, styrene, and smoke. Unlike the diffusion coefficient, the dispersion coefficient will be a strong function of position. It will have different values in different directions. Thus dispersion may look like diffusion, and it may be described by the same kinds of equations, but it is a different effect. 

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Perturbation theory yields a siim-over-states fomnila for each of the dispersion coefficients. For example, the isotropic coefficient for the interaction between molecules A and B is given by... 

The dispersion coefficient for interactions Q between molecules A and B can be estimated to an average... 

Thakkar A J, Hettema H and Wormer P E S 1992 Ab initio dispersion coefficients for interactions involving rare-gas atoms J. Chem. Phys. 97 3252... 

Thakkar A J 1988 Higher dispersion coefficients accurate values for hydrogen atoms and simple estimates for other systems J. Chem. Phys. 89 2092... 

Tang K T and Toennies J P 1984 An improved simple model for the van der Waals potential based on universal damping functions for the dispersion coefficients J. Chem. Phys. 80 3726... 

The eddy dispersion coefficient has been measured and correlated empirically as... 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Continuous stirred tank reactor Dispersion coefficient Effective diffusivity Knudsen diffusivity Residence time distribution Normalized residence time distribution... 

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. 

The recommended correlation for the gas-phase axial-dispersion coefficient is given by Field and Davidson (loc. cit.) ... 

In these expressions, B = ZJd, Nps = dVp/EE, Np r = dVn/Eii, where d = some characteristic length such as dp for packed towers or T for spray towers. Ep and Er are the longitudinal dispersion coefficients, which must ultimately be deter-... 

The axial dispersion coefficient [cf. Eq. (16-51)] lumps together all mechanisms leading to axial mixing in packed beds. Thus, the axial dispersion coefficient must account not only for moleciilar diffusion and convec tive mixing but also for nonuniformities in the fluid velocity across the packed bed. As such, the axial dispersion coefficient is best determined experimentally for each specific contac tor. 

Neglecting flow nonuniformities, the contributions of molecular diffusion and turbulent mixing arising from stream sphtting and recombination around the sorbent particles can be considered additive [Langer et al., Int. ]. Heat and Mass Transfer, 21, 751 (1978)] thus, the axial dispersion coefficient is given by ... 

FIG. 16 11 Axial dispersion coefficient correlations for well-packed beds for e = 0.4. 

A comparison of the axial-dispersion coefficients obtained in oscil-lating-spiral and dense-bed crystalhzers is given in Table 22-5. The dense-bed column approaches axial-dispersion coefficients similar to those of densely packed ice-washing cohimns. 

TABLE 22-5 Comparison of Axial-Dispersion Coefficients for Several Liquid-Solid Contactors... 

Column type Dispersion coefficient, cmVs Reference... 

J. Chem. Educ., 50, 864 (1973)], theory shows that the degree of separation that is obtained increases as the liquid column is made taller. But unfortunately it decreases as the column is made wider. In simple terms, the latter effect can be attributed to the increase in the dispersion coefficient as the column is widened. 

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, and so on. The effect is represented hy Tick s law with a dispersion coefficient substituted for molecular diffusivity. Thus, rate of transfer = —Dj3C/3p). 

Peclet number for dispersion Pe = uUD where t/ is a Bnear velocity, L is a hnear dimension, and is the dispersion coefficient. In packed beds, Pe = udp/De, where u is the interstitial velocity and dp is the pellet diameter. 

A related measure of efficiency is the equivalent number of stages erkngof CSTR battery with the same variance as the measured RTD. Practically, in some cases 5 or 6 stages may be taken to approximate plug flow. The dispersion coefficient also is a measure of deviation... 

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 dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

Re = R nolds number, dpS UolV Sc = Schmidt number, V/D D = axial dispersion coefficient dp = Diameter of particle or empty tube = Fraction voids in packed bed Uq = Superficial velocity in the vessel. 

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Two complementai y reviews of this subject are by Shah et al. AIChE Journal, 28, 353-379 [1982]) and Deckwer (in de Lasa, ed.. Chemical Reactor Design andTechnology, Martinus Nijhoff, 1985, pp. 411-461). Useful comments are made by Doraiswamy and Sharma (Heterogeneous Reactions, Wiley, 1984). Charpentier (in Gianetto and Silveston, eds.. Multiphase Chemical Reactors, Hemisphere, 1986, pp. 104—151) emphasizes parameters of trickle bed and stirred tank reactors. Recommendations based on the literature are made for several design parameters namely, bubble diameter and velocity of rise, gas holdup, interfacial area, mass-transfer coefficients k a and /cl but not /cg, axial liquid-phase dispersion coefficient, and heat-transfer coefficient to the wall. The effect of vessel diameter on these parameters is insignificant when D > 0.15 m (0.49 ft), except for the dispersion coefficient. Application of these correlations is to (1) chlorination of toluene in the presence of FeCl,3 catalyst, (2) absorption of SO9 in aqueous potassium carbonate with arsenite catalyst, and (3) reaction of butene with sulfuric acid to butanol. 

Plug flow is approached at low values of the dispersion coefficient or hi values or Peclet number. A criterion developed by Mears (Chem. Eng. Sci., 26, 1361 [1971]) is that conversion will be within 5 percent of that predicted by phig flow when... 

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