Dispersion longitudinal

consider the wakes downstream of the recirculation zone. White and Nepf [643]derived the following expression for the velocity deficit, uw, in the wake of a cylinder (stem) located at x = 0 and y = 0 within an array of cylinders, 

Velocity in the gaps is enhanced above the spatial average. Between adjacent cylinders located at x = 0 and y = w/2, where w is the gap width, the velocity may be 

Note that the dispersion terms described in equation (6.18) are valid only in the limit of Fickian behavior. From the central limit theorem, this regime is reached when every particle has amply sampled each region (wakes, gaps, recirculation zones). The average time-scale to advect through a wake is (a(u)Yl, and the average time-scale to experience trapping within a recirculation zone is r/ (yad). Then, the Fickian limit is reached at time t r/ (yad) and (fl(M 1. 

The sub-script h is used to denote the dispersion associated with velocity shear at the canopy scale, as distinct from velocity shear at the wake scale. Recall that here h = //, as the canopy is emergent. Equation (6.19) can be non-dimensionalized as 

Each term in (6.22) can be expressed in terms of the canopy parameters CDa, d, and h. Using equations (6.8) and (6.13), the non-dimensional dispersion is then, 

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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. 

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-... 

Giddings [2] estimated that, for a well-packed column, (y) takes a value of about 0.6. Equation (11) accurately describes longitudinal dispersion in GC capillary columns and equation (12) accurately describes longitudinal dispersion in GC and LC packed columns. Experimental support for these equations will be given in a later chapter. 

In summary, equation (13) accurately describes longitudinal dispersion in the stationary phase of capillary columns, but it will only be significant compared with other dispersion mechanisms in LC capillary columns, should they ever become generally practical and available. Dispersion due to longitudinal diffusion in the stationary phase in packed columns is not significant due to the discontinuous nature of the stationary phase and, compared to other dispersion processes, can be ignored in practice. 

Flow patterns in the reaetor ean vary greatly. To eharaeterize baekmixing, of the longitudinal dispersion number, D/uL, is often used,... 

This model is referred to as the axial dispersed plug flow model or the longitudinal dispersed plug flow model. (Dg)j. ean be negleeted relative to (Dg)[ when the ratio of eolumn diameter to length is very small and the flow is in the turbulent regime. This model is widely used for ehemieal reaetors and other eontaeting deviees. 

Axial and radial dispersion or non-ideal flow in tubular reactors is usually characterised by analogy to molecular diffusion, in which the molecular diffusivity is replaced by eddy dispersion coefficients, characterising both radial and longitudinal dispersion effects. In this text, however, the discussion will be limited to that of tubular reactors with axial dispersion only. Otherwise the model equations become too complicated and beyond the capability of a simple digital simulation language. 

We can characterize the mixed systems most easily in terms of the longitudinal dispersion model or in terms of the cascade of stirred tank reactors model. The maximum amount of mixing occurs for the cases where Q)L = oo or n = 1. In general, for reaction orders greater than unity, these models place a lower limit on the conversion that will be obtained in an actual reactor. The applications of these models are treated in Sections 11.2.2 and 11.2.3. 

The Longitudinal Dispersion Model in the Presence of Chemical Reaction... 

In Section 11.1.3.1 we considered the longitudinal dispersion model for flow in tubular reactors and indicated how one may employ tracer measurements to determine the magnitude of the dispersion parameter used in the model. In this section we will consider the problem of determining the conversion that will be attained when the model reactor operates at steady state. We will proceed by writing a material balance on a reactant species A using a tubular reactor. The mass balance over a reactor element of length AZ becomes ... 

Illustration 11.6 indicates how the longitudinal dispersion model may be used to predict reactor performance. 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

A more general one-dimensional model of tubular, packed bed reactors is contained within equations 12.7.38 and 12.7.47. These equations include all of the elements of the simple model discussed above and, in addition, account for the longitudinal dispersion of both thermal... 

In many respects, the solutions to equations 12.7.38 and 12.7.47 do not provide sufficient additional information to warrant their use in design calculations. It has been clearly demonstrated that for the fluid velocities used in industrial practice, the influence of axial dispersion of both heat and mass on the conversion achieved is negligible provided that the packing depth is in excess of 100 pellet diameters (109). Such shallow beds are only employed as the first stage of multibed adiabatic reactors. There is some question as to whether or not such short beds can be adequately described by an effective transport model. Thus for most preliminary design calculations, the simplified one-dimensional model discussed earlier is preferred. The discrepancies between model simulations and actual reactor behavior are not resolved by the inclusion of longitudinal dispersion terms. Their effects are small compared to the influence of radial gradients in temperature and composition. Consequently, for more accurate simulations, we employ a two-dimensional model (Section 12.7.2.2). 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Figure 16. Mixing coefficients for different vessel diameters Ml longitudinal dispersion of fluid, MB gas backmixing, Ms longitudinal solid dispersion. (From Van Deemter, 1980.)...

Longitudinal dispersion coefficients can be readily obtained by injecting a pulse of tracer into the bed in such a way that radial concentration gradients are eliminated, and measuring the change in shape of the pulse as it passes through the bed. Since dC/dr is then zero, equation 4.34 becomes ... 

Figure 4.6. Longitudinal dispersion in gases in packed beds. ER—Edwards and Richardson B — Blackwell etalS29 CB — Carberry and Bretton(32) DW—de Maria and White(33) MW—McHenry and Wilhelm(34) SP—Sinclair and Potter 35) EK — Evans and Kenney(36>, N2 + He...

However, the two mechanisms interact and molecular diffusion can reduce the effects of convective dispersion. This can be explained by the fact that with streamline flow in a tube molecular diffusion will tend to smooth out the concentration profile arising from the velocity distribution over the cross-section. Similarly radial dispersion can give rise to lower values of longitudinal dispersion than predicted by equation 4.39. As a result the curves of Peclet versus Reynolds number tend to pass through a maximum as shown in Figure 4.6. 

Figure 4.8. Longitudinal dispersion in liquids in packed beds. CP—Cairns and Prausnitz(37), CB — Carberry and Bretton(32) EW—Ebach and White(38) H — Hiby(39) LG—Liles and Geankoplis(40)...

Kramers et al. 25> have studied longitudinal dispersion in the liquid in a fluidised bed composed of glass spheres of 0.5 mm and 1 mm diameter. A step change was introduced... 

Kramers, H., Westermann, M. D., de Groot, 1. H. and Dupont, F. A. A. Third Congress of the European Federation of Chemical Engineering (1962). The Interaction between Fluids and Particles 114. The longitudinal dispersion of liquid in a fluidised bed. 

In liquids the effects of longitudinal dispersion are small, even at low Reynolds Numbers. 

The condition for which the bed is likely to operate at near equilibrium is when the feed rate is low. This is also the condition when longitudinal dispersion may be significant. Equilibrium solutions have been found by Lapidus and Amundson(39) and by Levenspiel and liisnioi i 40 for this case. These take the form ... 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

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